Research Article Weyl-Euler-Lagrange Equations of...
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Research Article Weyl-Euler-Lagrange Equations of Motion on Flat Manifold Zeki Kasap Department of Elementary Education, Faculty of Education, Pamukkale University, Kinikli Campus, Denizli, Turkey Correspondence should be addressed to Zeki Kasap; [email protected] Received 27 April 2015; Accepted 11 May 2015 Academic Editor: John D. Clayton Copyright © 2015 Zeki Kasap. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper deals with Weyl-Euler-Lagrange equations of motion on flat manifold. It is well known that a Riemannian manifold is said to be flat if its curvature is everywhere zero. Furthermore, a flat manifold is one Euclidean space in terms of distances. Weyl introduced a metric with a conformal transformation for unified theory in 1918. Classical mechanics is one of the major subfields of mechanics. Also, one way of solving problems in classical mechanics occurs with the help of the Euler-Lagrange equations. In this study, partial differential equations have been obtained for movement of objects in space and solutions of these equations have been generated by using the symbolic Algebra soſtware. Additionally, the improvements, obtained in this study, will be presented. 1. Introduction Euler-Lagrangian (analogues) mechanics are very important tools for differential geometry and analytical mechanics. ey have a simple method to describe the model for mechanical systems. e models for mechanical systems are related. Studies in the literature about the Weyl manifolds are given as follows. Liu and Jun expand electronic origins, molecular dynamics simulations, computational nanomechanics, and multiscale modelling of materials fields [1]. Tekkoyun and Yayli examined generalized-quaternionic K¨ ahlerian analogue of Lagrangian and Hamiltonian mechanical systems [2]. e study given in [3] has the particular purpose to examine the discussion Weyl and Einstein had over Weyl’s 1918 uni- fied field theory for reasons such as the epistemological implications. Kasap and Tekkoyun investigated Lagrangian and Hamiltonian formalism for mechanical systems using para-/pseudo-K¨ ahler manifolds, representing an interesting multidisciplinary field of research [4]. Kasap obtained the Weyl-Euler-Lagrange and the Weyl-Hamilton equations on R 2 which is a model of tangent manifolds of constant - sectional curvature [5]. Kapovich demonstrated an existence theorem for flat conformal structures on finite-sheeted cov- erings over a wide class of Haken manifolds [6]. Schwartz accepted asymptotically Riemannian manifolds with non- negative scalar curvature [7]. Kulkarni identified some new examples of conformally flat manifolds [8]. Dotti and Miatello intend to find out the real cohomology ring of low dimensional compact flat manifolds endowed with one of these special structures [9]. Szczepanski presented a list of six- dimensional K¨ ahler manifolds and he submitted an example of eight-dimensional K¨ ahler manifold with finite group [10]. Bartnik showed that the mass of an asymptotically flat - manifold is a geometric invariant [11]. Gonz´ alez considered complete, locally conformally flat metrics defined on a domain Ω⊂ [12]. Akbulut and Kalafat established infinite families of nonsimply connected locally conformally flat (LCF) 4-manifold realizing rich topological types [13]. Zhu suggested that it is to give a classification of complete locally conformally flat manifolds of nonnegative Ricci curvature [14]. Abood studied this tensor on general class almost Her- mitian manifold by using a new methodology which is called an adjoint -structure space [15]. K. Olszak and Z. Olszak proposed paraquaternionic analogy of these ideas applied to conformally flat almost pseudo-K¨ ahlerian as well as almost para-K¨ ahlerian manifolds [16]. Upadhyay studied bounding question for almost manifolds by looking at the equivalent description of them as infranil manifolds Γ \ ⋊ / [17]. 2. Preliminaries Definition 1. With respect to tangent space given any point ∈, it has a tangent space isometric to R . If one has a metric (inner-product) in this space ⟨, ⟩ : × → Hindawi Publishing Corporation Advances in Mathematical Physics Volume 2015, Article ID 808016, 11 pages http://dx.doi.org/10.1155/2015/808016
Transcript of Research Article Weyl-Euler-Lagrange Equations of...
Research ArticleWeyl-Euler-Lagrange Equations of Motion on Flat Manifold
Zeki Kasap
Department of Elementary Education Faculty of Education Pamukkale University Kinikli Campus Denizli Turkey
Correspondence should be addressed to Zeki Kasap zekikasaphotmailcom
Received 27 April 2015 Accepted 11 May 2015
Academic Editor John D Clayton
Copyright copy 2015 Zeki Kasap This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper deals with Weyl-Euler-Lagrange equations of motion on flat manifold It is well known that a Riemannian manifold issaid to be flat if its curvature is everywhere zero Furthermore a flat manifold is one Euclidean space in terms of distances Weylintroduced a metric with a conformal transformation for unified theory in 1918 Classical mechanics is one of the major subfieldsof mechanics Also one way of solving problems in classical mechanics occurs with the help of the Euler-Lagrange equations Inthis study partial differential equations have been obtained for movement of objects in space and solutions of these equations havebeen generated by using the symbolic Algebra software Additionally the improvements obtained in this study will be presented
1 Introduction
Euler-Lagrangian (analogues) mechanics are very importanttools for differential geometry and analyticalmechanicsTheyhave a simple method to describe the model for mechanicalsystems The models for mechanical systems are relatedStudies in the literature about the Weyl manifolds are givenas follows Liu and Jun expand electronic origins moleculardynamics simulations computational nanomechanics andmultiscale modelling of materials fields [1] Tekkoyun andYayli examined generalized-quaternionic Kahlerian analogueof Lagrangian and Hamiltonian mechanical systems [2] Thestudy given in [3] has the particular purpose to examinethe discussion Weyl and Einstein had over Weylrsquos 1918 uni-fied field theory for reasons such as the epistemologicalimplications Kasap and Tekkoyun investigated Lagrangianand Hamiltonian formalism for mechanical systems usingpara-pseudo-Kahler manifolds representing an interestingmultidisciplinary field of research [4] Kasap obtained theWeyl-Euler-Lagrange and the Weyl-Hamilton equations onR2119899119899
which is a model of tangent manifolds of constant 119882-sectional curvature [5] Kapovich demonstrated an existencetheorem for flat conformal structures on finite-sheeted cov-erings over a wide class of Haken manifolds [6] Schwartzaccepted asymptotically Riemannian manifolds with non-negative scalar curvature [7] Kulkarni identified somenew examples of conformally flat manifolds [8] Dotti and
Miatello intend to find out the real cohomology ring of lowdimensional compact flat manifolds endowed with one ofthese special structures [9] Szczepanski presented a list of six-dimensional Kahler manifolds and he submitted an exampleof eight-dimensional Kahler manifold with finite group [10]Bartnik showed that the mass of an asymptotically flat 119899-manifold is a geometric invariant [11] Gonzalez consideredcomplete locally conformally flat metrics defined on adomain Ω sub 119878
119899 [12] Akbulut and Kalafat established infinitefamilies of nonsimply connected locally conformally flat(LCF) 4-manifold realizing rich topological types [13] Zhusuggested that it is to give a classification of complete locallyconformally flat manifolds of nonnegative Ricci curvature[14] Abood studied this tensor on general class almost Her-mitian manifold by using a newmethodology which is calledan adjoint 119866-structure space [15] K Olszak and Z Olszakproposed paraquaternionic analogy of these ideas applied toconformally flat almost pseudo-Kahlerian as well as almostpara-Kahlerian manifolds [16] Upadhyay studied boundingquestion for almost manifolds by looking at the equivalentdescription of them as infranil manifolds Γ 119871 ⋊ 119866119866 [17]
2 Preliminaries
Definition 1 With respect to tangent space given any point119901 isin 119872 it has a tangent space 119879
119901119872 isometric toR119899 If one has
a metric (inner-product) in this space ⟨ ⟩119901 119879119901119872times 119879
119901119872 997891rarr
Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2015 Article ID 808016 11 pageshttpdxdoiorg1011552015808016
2 Advances in Mathematical Physics
R defined on every point 119901 isin 119872119872 is called a Riemannianmanifold
Definition 2 A manifold with a Riemannian metric is a flatmanifold such that it has zero curvature
Definition 3 A differentiable manifold 119872 is said to be analmost complex manifold if there exists a linear map 119869
119879119872 rarr 119879119872 satisfying 1198692 = minus119894119889 and 119869 is said to be an almostcomplex structure of119872 where 119894 is the identity (unit) operatoron 119881 such that 119881 is the vector space and 1198692 = 119869 ∘ 119869
Theorem 4 The integrability of the almost complex structureimplies a relation in the curvature Let 1199091 1199101 1199092 1199102 1199093 1199103 becoordinates on R6 with the standard flat metric
1198891199042=
3sum
119894=1(119889119909
2119894+119889119910
2119894) (1)
(see [18])
Definition 5 A (pseudo-)Riemannian manifold is confor-mally flat manifold if each point has a neighborhood that canbe mapped to flat space by a conformal transformation Let(119872 119892) be a pseudo-Riemannian manifold
Theorem 6 Let (119872 119892) be conformally flat if for each point 119909in119872 there exists a neighborhood119880 of 119909 and a smooth function119891 defined on 119880 such that (119880 1198902119891119892) is flat The function 119891 neednot be defined on all of119872 Some authors use locally conformallyflat to describe the above notion and reserve conformally flat forthe case in which the function 119891 is defined on all of119872 [19]
Definition 7 A pseudo-119869-holomorphic curve is a smoothmap from a Riemannian surface into an almost complexmanifold such that it satisfies the Cauchy-Riemann equation[20]
Definition 8 A conformal map is a function which preservesangles as the most common case where the function isbetween domains in the complex plane Conformal maps canbe defined betweendomains in higher dimensional Euclideanspaces andmore generally on a (semi-)Riemannianmanifold
Definition 9 Conformal geometry is the study of the setof angle-preserving (conformal) transformations on a spaceIn two real dimensions conformal geometry is preciselythe geometry of Riemannian surfaces In more than twodimensions conformal geometry may refer either to thestudy of conformal transformations of flat spaces (such asEuclidean spaces or spheres) or to the study of conformalmanifolds which are Riemannian or pseudo-Riemannianmanifolds with a class of metrics defined up to scale
Definition 10 A conformal manifold is a differentiable mani-fold equippedwith an equivalence class of (pseudo-)Riemannmetric tensors in which two metrics 1198921015840 and 119892 are equivalentif and only if
1198921015840
= Ψ2119892 (2)
where Ψ gt 0 is a smooth positive function An equivalenceclass of such metrics is known as a conformal metric orconformal class and a manifold with a conformal structureis called a conformal manifold [21]
3 Weyl Geometry
Conformal transformation for use in curved lengths has beenrevealed The linear distance between two points can befound easily by Riemann metric Many scientists have usedthe Riemann metric Einstein was one of the first to studythis field Einstein discovered the Riemannian geometry andsuccessfully used it to describe general relativity in the 1910that is actually a classical theory for gravitation But theuniverse is really completely not like Riemannian geometryEach path between two points is not always linear Alsoorbits of moving objects may change during movement Soeach two points in space may not be linear geodesic Thena method is required for converting nonlinear distance tolinear distance Weyl introduced a metric with a conformaltransformation in 1918The basic concepts related to the topicare listed below [22ndash24]
Definition 11 Two Riemann metrics 1198921 and 1198922 on119872 are saidto be conformally equivalent iff there exists a smooth function119891 119872 rarr R with
119890119891
1198921 = 1198922 (3)
In this case 1198921 sim 1198922
Definition 12 Let119872 be an 119899-dimensional smooth manifoldA pair (119872119866) where a conformal structure on 119872 is anequivalence class 119866 of Riemann metrics on 119872 is called aconformal structure
Theorem 13 Let nabla be a connection on119872 and 119892 isin 119866 a fixedmetric nabla is compatible with (119872119866) hArr there exists a 1-form 120596
with nabla119883119892 + 120596(119883)119892 = 0
Definition 14 A compatible torsion-free connection is calleda Weyl connection The triple (119872119866nabla) is a Weyl structure
Theorem 15 To each metric 119892 isin 119866 and 1-form 120596 there corre-sponds a unique Weyl connection nabla satisfying nabla
119883119892 +120596(119883)119892 =
0
Definition 16 Define a function 119865 1-forms on 119872 times 119866 rarr
Weyl connections by 119865(119892 120596) = nabla where nabla is the connec-tion guaranteed by Theorem 6 One says that nabla correspondsto (119892 120596)
Proposition 17 (1) 119865 is surjective
Proof 119865 is surjective byTheorem 13
(2) 119865(119892 120596) = 119865(119890119891119892 120578) iff 120578 = 120596 minus 119889119891 So
119865 (119890119891
119892) = 119865 (119892) minus 119889119891 (4)
where 119866 is a conformal structure Note that a Riemann metric119892 and a one-form 120596 determine a Weyl structure namely 119865
Advances in Mathematical Physics 3
119866 rarr and1119872 where119866 is the equivalence class of 119892 and119865(119890119891119892) =
120596 minus 119889119891
Proof Suppose that 119865(119892 120596) = 119865(119890119891119892 120578) = nabla We have
nabla119883(119890119891
119892) + 120578 (119883) 119890119891
119892
= 119883(119890119891
) 119892 + 119890119891
nabla119883119892+ 120578 (119883) 119890
119891
119892
= 119889119891 (119883) 119890119891
119892+ 119890119891
nabla119883119892+ 120578 (119883) 119890
119891
119892 = 0
(5)
Thereforenabla119883119892 = minus(119889119891(119883)+120578(119883)) On the other handnabla
119883119892+
120596(119883)119892 = 0Therefore 120596 = 120578 + 119889119891 Set nabla = 119865(119892 120596) To shownabla = 119865(119890
119891
119892 120578) and nabla119883(119890119891
119892) + 120578(119883)119890119891
119892 = 0 To calculate
nabla119883(119890119891
119892) + 120578 (119883) 119890119891
119892
= 119890119891
119889119891 (119883) 119892 + 119890119891
nabla119883119892+ (120596 (119883) minus 119889119891 (119883)) 119890
119891
119892
= 119890119891
(nabla119883119892+120596 (119883) 119892) = 0
(6)
Theorem 18 A connection on the metric bundle 120596 of aconformalmanifold119872naturally induces amap119865 119866 rarr and
1119872
and (4) and conversely Parallel translation of points in120596 by theconnection is the same as their translation by 119865
Theorem 19 Let nabla be a torsion-free connection on the tangentbundle of 119872 and 119898 ge 6 If (119872 119892 nabla 119869) is a Kahler-Weylstructure then the associated Weyl structure is trivial that isthere is a conformally equivalent metric 119892 = 119890
2119891119892 so that
(119872 119892 119869) is Kahler and so that nabla = nabla119892 [25ndash27]
Definition 20 Weyl curvature tensor is a measure of thecurvature of spacetime or a pseudo-Riemannian manifoldLike the Riemannian curvature tensor the Weyl tensorexpresses the tidal force that a body feels when moving alonga geodesic
Definition 21 Weyl transformation is a local rescaling ofthe metric tensor 119892
119886119887(119909) rarr 119890
minus2120596(119909)
119892119886119887(119909) which produces
another metric in the same conformal class A theory oran expression invariant under this transformation is calledconformally invariant or is said to possess Weyl symmetryTheWeyl symmetry is an important symmetry in conformalfield theory
4 Complex Structures on ConformallyFlat Manifold
In this sectionWeyl structures on flatmanifoldswill be trans-ferred to the mechanical system Thus the time-dependentEuler-Lagrange partial equations of motion of the dynamicsystemwill be found A flatmanifold is something that locallylooks like Euclidean space in terms of distances and anglesThe basic example is Euclidean space with the usual metric119889119904
2= sum119894119889119909
2119894 Any point on a flat manifold has a neighbor-
hood isometric to a neighborhood in Euclidean space A flatmanifold is locally Euclidean in terms of distances and anglesand merely topologically locally Euclidean as all manifolds
are The simplest nontrivial examples occur as surfaces infour-dimensional space as the flat torus is a flat manifold Itis the image of 119891(119909 119910) = (cos119909 sin119909 cos119910 sin119910)
Example 22 It vanishes if and only if 119869 is an integrable almostcomplex structure that is given any point 119875 isin 119872 there existlocal coordinates (119909
119894 119910119894) 119894 = 1 2 3 centered at 119875 following
structures taken from
1198691205971199091 = cos (1199093) 1205971199101 + sin (1199093) 1205971199102
1198691205971199092 = minus sin (1199093) 1205971199101 + cos (1199093) 1205971199102
1198691205971199093 = 1205971199103
1198691205971199101 = minus cos (1199093) 1205971199091 + sin (1199093) 1205971199092
1198691205971199102 = minus sin (1199093) 1205971199091 minus cos (1199093) 1205971199092
1198691205971199103 = minus 1205971199093
(7)
The above structures (7) have been taken from [28] We willuse 120597119909
119894= 120597120597119909
119894and 120597119910
119894= 120597120597119910
119894
The Weyl tensor differs from the Riemannian curvaturetensor in that it does not convey information on how thevolume of the body changes In dimensions 2 and 3 theWeyl curvature tensor vanishes identically Also the Weylcurvature is generally nonzero for dimensions ge4 If theWeyltensor vanishes in dimension ge4 then the metric is locallyconformally flat there exists a local coordinate system inwhich the metric tensor is proportional to a constant tensorThis fact was a key component for gravitation and generalrelativity [29]
Proposition 23 If we extend (7) by means of conformalstructure [19 30] Theorem 19 and Definition 21 we can giveequations as follows
119869120597
1205971199091= 119890
2119891 cos (1199093)120597
1205971199101+ 119890
2119891 sin (1199093)120597
1205971199102
119869120597
1205971199092= minus 119890
2119891 sin (1199093)120597
1205971199101+ 119890
2119891 cos (1199093)120597
1205971199102
119869120597
1205971199093= 119890
2119891 120597
1205971199103
119869120597
1205971199101= minus 119890minus2119891 cos (1199093)
120597
1205971199091+ 119890minus2119891 sin (1199093)
120597
1205971199092
119869120597
1205971199102= minus 119890minus2119891 sin (1199093)
120597
1205971199091minus 119890minus2119891 cos (1199093)
120597
1205971199092
119869120597
1205971199103= minus 119890minus2119891 120597
1205971199093
(8)
such that they are base structures for Weyl-Euler-Lagrangeequations where 119869 is a conformal complex structure to be simi-lar to an integrable almost complex 119869 given in (7) Fromnow onwe continue our studies thinking of the (119879119872 119892 nabla 119869) instead of
4 Advances in Mathematical Physics
Weyl manifolds (119879119872 119892 nabla 119869) Now 119869 denotes the structure ofthe holomorphic property
1198692 120597
1205971199091= 119869 ∘ 119869
120597
1205971199091= 119890
2119891 cos (1199093) 119869120597
1205971199101+ 119890
2119891 sin (1199093) 119869
sdot120597
1205971199102= 119890
2119891 cos (1199093)
sdot [minus119890minus2119891 cos (1199093)
120597
1205971199091+ 119890minus2119891 sin (1199093)
120597
1205971199092]+ 119890
2119891
sdot sin (1199093) [minus119890minus2119891 sin (1199093)
120597
1205971199091minus 119890minus2119891 cos (1199093)
120597
1205971199092]
= minus cos2 (1199093)120597
1205971199091+ cos (1199093) sin (1199093)
120597
1205971199092
minus sin2 (1199093)120597
1205971199091minus sin (1199093) cos (1199093)
120597
1205971199092
= minus [cos2 (1199093) + sin2(1199093)]
120597
1205971199091= minus
120597
1205971199091
(9)
and in similar manner it is shown that
1198692 120597
120597119909119894
= minus120597
120597119909119894
1198692 120597
120597119910119894
= minus120597
120597119910119894
119894 = 1 2 3
(10)
As can be seen from (9) and (10) 1198692 = minus119868 are the complexstructures
5 Euler-Lagrange Dynamics Equations
Definition 24 (see [31ndash33]) Let119872 be an 119899-dimensional man-ifold and 119879119872 its tangent bundle with canonical projection120591119872 119879119872 rarr 119872 119879119872 is called the phase space of velocities of
the base manifold119872 Let 119871 119879119872 rarr R be a differentiablefunction on 119879119872 and it is called the Lagrangian function Weconsider closed 2-form on 119879119872 and Φ
119871= minus119889d
119869119871 Consider
the equation
i119881Φ119871= 119889119864119871 (11)
where the semispray 119881 is a vector field Also i is a reducingfunction and i
119881Φ119871= Φ119871(119881) We will see that for motion in
a potential 119864119871= V(119871) minus 119871 is an energy function (119871 = 119879minus119875 =
(12)119898V2 minus 119898119892ℎ kinetic-potential energies) and V = 119869119881
a Liouville vector field Here 119889119864119871denotes the differential
of 119864 We will see that (11) under a certain condition on 119881is the intrinsic expression of the Euler-Lagrange equationsof motion This equation is named Euler-Lagrange dynam-ical equation The triple (119879119872Φ
119871 119881) is known as Euler-
Lagrangian systemon the tangent bundle119879119872Theoperationsrun on (11) for any coordinate system (119902
119894
(119905) 119901119894(119905)) Infinite
dimension Lagrangianrsquos equation is obtained in the formbelow
119889
119889119905(120597119871
120597 119902119894)minus
120597119871
120597119902119894= 0
119889119902119894
119889119905= 119902119894
119894 = 1 119899
(12)
6 Conformal Weyl-Euler-LagrangianEquations
Here we using (11) obtain Weyl-Euler-Lagrange equationsfor classical and quantum mechanics on conformally flatmanifold and it is shown by (119879119872 119892 nabla 119869)
Proposition 25 Let (119909119894 119910119894) be coordinate functions Also on
(119879119872 119892 nabla 119869) let 119881 be the vector field determined by 119881 =
sum3119894=1(119883119894
(120597120597119909119894) + 119884119894
(120597120597119910119894)) Then the vector field defined by
V = 119869119881
= 1198831(119890
2119891 cos (1199093)120597
1205971199101+ 119890
2119891 sin (1199093)120597
1205971199102)
+1198832(minus119890
2119891 sin (1199093)120597
1205971199101+ 119890
2119891 cos (1199093)120597
1205971199102)
+11988331198902119891 120597
1205971199103
+1198841(minus119890minus2119891 cos (1199093)
120597
1205971199091+ 119890minus2119891 sin (1199093)
120597
1205971199092)
+1198842(minus119890minus2119891 sin (1199093)
120597
1205971199091minus 119890minus2119891 cos (1199093)
120597
1205971199092)
minus1198843119890minus2119891 120597
1205971199093
(13)
is thought to be Weyl-Liouville vector field on conformally flatmanifold (119879119872 119892 nabla 119869) Φ
119871= minus119889d
119869119871 is the closed 2-form
given by (11) such that d = sum3119894=1((120597120597119909119894)119889119909119894 + (120597120597119910119894)119889119910119894)
d119869 119865(119872) rarr and
1119872 d119869= 119894119869d minus d119894
119869 and d
119869= 119869(d) =
sum3119894=1(119883119894
119869(120597120597119909119894)+119884119894
119869(120597120597119910119894)) Also the vertical differentiation
d119869is given where 119889 is the usual exterior derivationThen there
is the following result We can obtain Weyl-Euler-Lagrangeequations for classical and quantummechanics on conformallyflat manifold (119879119872 119892 nabla 119869) We get the equations given by
d119869= [119890
2119891 cos (1199093)120597
1205971199101+ 119890
2119891 sin (1199093)120597
1205971199102] 1198891199091
+[minus1198902119891 sin (1199093)
120597
1205971199101+ 119890
2119891 cos (1199093)120597
1205971199102] 1198891199092
+ 1198902119891 120597
12059711991031198891199093
Advances in Mathematical Physics 5
+[minus119890minus2119891 cos (1199093)
120597
1205971199091+ 119890minus2119891 sin (1199093)
120597
1205971199092] 1198891199101
+[minus119890minus2119891 sin (1199093)
120597
1205971199091minus 119890minus2119891 cos (1199093)
120597
1205971199092] 1198891199102
minus 119890minus2119891 120597
12059711990931198891199103
(14)
Also
Φ119871= minus119889d
119869119871
= minus119889([1198902119891 cos (1199093)
120597
1205971199101+ 119890
2119891 sin (1199093)120597
1205971199102] 1198891199091
+[minus1198902119891 sin (1199093)
120597
1205971199101+ 119890
2119891 cos (1199093)120597
1205971199102] 1198891199092
+ 1198902119891 120597119871
12059711991031198891199093
+[minus119890minus2119891 cos (1199093)
120597
1205971199091+ 119890minus2119891 sin (1199093)
120597
1205971199092] 1198891199101
+[minus119890minus2119891 sin (1199093)
120597
1205971199091minus 119890minus2119891 cos (1199093)
120597
1205971199092] 1198891199102
minus 119890minus2119891 120597119871
12059711990931198891199103)
(15)
and then we find
i119881Φ119871= Φ119871(119881) = Φ
119871(
3sum
119894=1(119883119894120597
120597119909119894
+119884119894120597
120597119910119894
)) (16)
Moreover the energy function of system is
119864119871= 119883
1[119890
2119891 cos (1199093)120597119871
1205971199101+ 119890
2119891 sin (1199093)120597119871
1205971199102]
+1198832[minus119890
2119891 sin (1199093)120597119871
1205971199101+ 119890
2119891 cos (1199093)120597119871
1205971199102]
+11988331198902119891 120597119871
1205971199103
+1198841[minus119890minus2119891 cos (1199093)
120597119871
1205971199091+ 119890minus2119891 sin (1199093)
120597119871
1205971199092]
+1198842[minus119890minus2119891 sin (1199093)
120597119871
1205971199091minus 119890minus2119891 cos (1199093)
120597119871
1205971199092]
minus1198843119890minus2119891 120597119871
1205971199093minus119871
(17)
and the differential of 119864119871is
119889119864119871= 119883
1(119890
2119891 cos (1199093)1205972119871
120597119909112059711991011198891199091
+ 21198902119891 cos (1199093)120597119891
1205971199091
120597119871
12059711991011198891199091
+ 1198902119891 sin (1199093)
1205972119871
120597119909112059711991021198891199091
+ 21198902119891 sin (1199093)120597119891
1205971199091
120597119871
12059711991021198891199091)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119909112059711991011198891199091
minus 21198902119891 sin (1199093)120597119891
1205971199091
120597119871
12059711991011198891199091
+ 1198902119891 cos (1199093)
1205972119871
120597119909112059711991021198891199091
+ 21198902119891 cos (1199093)120597119891
1205971199091
120597119871
12059711991021198891199091)
+1198833(119890
2119891 1205972119871
120597119909112059711991031198891199091 + 2119890
2119891 120597119891
1205971199091
120597119871
12059711991031198891199091)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119909211198891199091
+ 2119890minus2119891 cos (1199093)120597119891
1205971199091
120597119871
12059711990911198891199091
+ 119890minus2119891 sin (1199093)
1205972119871
120597119909112059711990921198891199091
minus 2119890minus2119891 sin (1199093)120597119891
1205971199091
120597119871
12059711990921198891199091)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119909211198891199091
+ 2119890minus2119891 sin (1199093)120597119891
1205971199091
120597119871
12059711990911198891199091
minus 119890minus2119891 cos (1199093)
1205972119871
120597119909112059711990921198891199091
+ 2119890minus2119891 cos (1199093)120597119891
1205971199091
120597119871
12059711990921198891199091)
+1198843(minus
1205972119871
120597119909112059711990931198891199091 + 2119890
minus2119891 120597119891
1205971199091
120597
12059711990931198891199091)
minus120597119871
12059711990911198891199091 +119883
1(119890
2119891 cos (1199093)1205972119871
120597119909212059711991011198891199092
6 Advances in Mathematical Physics
+ 21198902119891 cos (1199093)120597119891
1205971199092
120597119871
12059711991011198891199092
+ 1198902119891 sin (1199093)
1205972119871
120597119909212059711991021198891199092
+ 21198902119891 sin (1199093)120597119891
1205971199092
120597119871
12059711991021198891199092)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119909212059711991011198891199092
minus 21198902119891 sin (1199093)120597119891
1205971199092
120597119871
12059711991011198891199092
+ 1198902119891 cos (1199093)
1205972119871
120597119909212059711991021198891199092
+ 21198902119891 cos (1199093)120597119891
1205971199092
120597119871
12059711991021198891199092)
+1198833(119890
2119891 1205972119871
120597119909212059711991031198891199092 + 2119890
2119891 120597119891
1205971199092
120597119871
12059711991031198891199092)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119909212059711990911198891199092
+ 2119890minus2119891 cos (1199093)120597119891
1205971199092
120597119871
12059711990911198891199092
+ 119890minus2119891 sin (1199093)
1205972119871
120597119909221198891199092
minus 2119890minus2119891 sin (1199093)120597119891
1205971199092
120597119871
12059711990921198891199092)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119909212059711990911198891199092
+ 2119890minus2119891 sin (1199093)120597119891
1205971199092
120597119871
12059711990911198891199092
minus 119890minus2119891 cos (1199093)
1205972119871
120597119909221198891199092
+ 2119890minus2119891 cos (1199093)120597119891
1205971199092
120597119871
12059711990921198891199092)
+1198843(minus
1205972119871
120597119909212059711990931198891199092 + 2119890
minus2119891 120597119891
1205971199092
120597
12059711990931198891199092)
minus120597119871
12059711990921198891199092 +119883
1(119890
2119891 cos (1199093)1205972119871
120597119909312059711991011198891199093
+ 21198902119891 cos (1199093)120597119891
1205971199093
120597119871
12059711991011198891199093 minus 119890
2119891 sin (1199093)120597119871
12059711991011198891199093
+ 1198902119891 sin (1199093)
1205972119871
120597119909312059711991021198891199093
+ 21198902119891 sin (1199093)120597119891
1205971199093
120597119871
12059711991021198891199093
+ 1198902119891 cos (1199093)
120597119871
12059711991021198891199093)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119909312059711991011198891199093
minus 21198902119891 sin (1199093)120597119891
1205971199093
120597119871
12059711991011198891199093 minus 119890
2119891 cos (1199093)120597119871
12059711991011198891199093
+ 1198902119891 cos (1199093)
1205972119871
120597119909312059711991021198891199093
+ 21198902119891 cos (1199093)120597119891
1205971199093
120597119871
12059711991021198891199093
minus 1198902119891 sin (1199093)
120597119871
12059711991021198891199093)+119883
3(119890
2119891 1205972119871
120597119909312059711991031198891199093
+ 21198902119891120597119891
1205971199093
120597119871
12059711991031198891199093)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119909312059711990911198891199093
+ 2119890minus2119891 cos (1199093)120597119891
1205971199093
120597119871
12059711990911198891199093
minus 119890minus2119891 sin (1199093)
120597119871
12059711990911198891199093 + 119890
minus2119891 sin (1199093)1205972119871
120597119909312059711990921198891199093
minus 2119890minus2119891 sin (1199093)120597119891
1205971199093
120597119871
12059711990921198891199093
+ 119890minus2119891 cos (1199093)
120597119871
12059711990921198891199093)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119909312059711990911198891199093
+ 2119890minus2119891 sin (1199093)120597119891
1205971199093
120597119871
12059711990911198891199093
minus 119890minus2119891 cos (1199093)
120597119871
12059711990911198891199093
minus 119890minus2119891 cos (1199093)
1205972119871
120597119909312059711990921198891199093
+ 2119890minus2119891 cos (1199093)120597119891
1205971199093
120597119871
12059711990921198891199093
+ 119890minus2119891 sin (1199093)
120597119871
12059711990921198891199093)+119884
3(minus
1205972119871
120597119909231198891199093
+ 2119890minus2119891120597119891
1205971199093
120597
12059711990931198891199093)minus
120597119871
12059711990931198891199093
Advances in Mathematical Physics 7
+1198831(119890
2119891 cos (1199093)1205972119871
120597119910211198891199101
+ 21198902119891 cos (1199093)120597119891
1205971199101
120597119871
12059711991011198891199101
+ 1198902119891 sin (1199093)
1205972119871
120597119910112059711991021198891199101
+ 21198902119891 sin (1199093)120597119891
1205971199101
120597119871
12059711991021198891199101)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119910211198891199101
minus 21198902119891 sin (1199093)120597119891
1205971199101
120597119871
12059711991011198891199101
+ 1198902119891 cos (1199093)
1205972119871
120597119910112059711991021198891199101
+ 21198902119891 cos (1199093)120597119891
1199101
120597119871
12059711991021198891199101)+119883
3(119890
2119891 1205972119871
120597119910112059711991031198891199101
+ 21198902119891120597119891
1205971199101
120597119871
12059711991031198891199101)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119910112059711990911198891199101
+ 2119890minus2119891 cos (1199093)120597119891
1205971199101
120597119871
12059711990911198891199101
+ 119890minus2119891 sin (1199093)
1205972119871
120597119910112059711990921198891199101
minus 2119890minus2119891 sin (1199093)120597119891
1205971199101
120597119871
12059711990921198891199101)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119910112059711990911198891199101
+ 2119890minus2119891 sin (1199093)120597119891
1205971199101
120597119871
12059711990911198891199101
minus 119890minus2119891 cos (1199093)
1205972119871
120597119910112059711990921198891199101
+ 2119890minus2119891 cos (1199093)120597119891
1205971199101
120597119871
12059711990921198891199101)+119884
3(minus
1205972119871
120597119910112059711990931198891199101
+ 2119890minus2119891120597119891
1205971199101
120597
12059711990931198891199101)minus
120597119871
12059711991011198891199101
+1198831(119890
2119891 cos (1199093)1205972119871
120597119910212059711991011198891199102
+ 21198902119891 cos (1199093)120597119891
1205971199102
120597119871
12059711991011198891199102 + 119890
2119891 sin (1199093)1205972119871
120597119910221198891199102
+ 21198902119891 sin (1199093)120597119891
1205971199102
120597119871
12059711991021198891199102)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119910212059711991011198891199102
minus 21198902119891 sin (1199093)120597119891
1205971199102
120597119871
12059711991011198891199102 + 119890
2119891 cos (1199093)1205972119871
120597119910221198891199102
+ 21198902119891 cos (1199093)120597119891
1205971199102
120597119871
12059711991021198891199102)
+1198833(119890
2119891 1205972119871
120597119910212059711991031198891199102 + 2119890
2119891 120597119891
1205971199102
120597119871
12059711991031198891199102)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119910212059711990911198891199102
+ 2119890minus2119891 cos (1199093)120597119891
1205971199102
120597119871
12059711990911198891199102
+ 119890minus2119891 sin (1199093)
1205972119871
120597119910212059711990921198891199102
minus 2119890minus2119891 sin (1199093)120597119891
1205971199102
120597119871
12059711990921198891199102)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119910212059711990911198891199102
+ 2119890minus2119891 sin (1199093)120597119891
1205971199102
120597119871
12059711990911198891199102
minus 119890minus2119891 cos (1199093)
1205972119871
120597119910212059711990921198891199102
+ 2119890minus2119891 cos (1199093)120597119891
1205971199102
120597119871
12059711990921198891199102)+119884
3(minus
1205972119871
120597119910212059711990931198891199102
+ 2119890minus2119891120597119891
1205971199102
120597
12059711990931198891199102)minus
120597119871
12059711991021198891199102
+1198831(119890
2119891 cos (1199093)1205972119871
120597119910312059711991011198891199103
+ 21198902119891 cos (1199093)120597119891
1205971199103
120597119871
12059711991011198891199103
+ 1198902119891 sin (1199093)
1205972119871
120597119910312059711991021198891199103
+ 21198902119891 sin (1199093)120597119891
1205971199103
120597119871
12059711991021198891199103)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119910312059711991011198891199103
minus 21198902119891 sin (1199093)120597119891
1205971199103
120597119871
12059711991011198891199103
8 Advances in Mathematical Physics
+ 1198902119891 cos (1199093)
1205972119871
120597119910312059711991021198891199103
+ 21198902119891 cos (1199093)120597119891
1205971199103
120597119871
12059711991021198891199103)+119883
3(119890
2119891 1205972119871
120597119910231198891199103
+ 21198902119891120597119891
1205971199103
120597119871
12059711991031198891199103)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119910312059711990911198891199103
+ 2119890minus2119891 cos (1199093)120597119891
1205971199103
120597119871
12059711990911198891199103
+ 119890minus2119891 sin (1199093)
1205972119871
120597119910312059711990921198891199103
minus 2119890minus2119891 sin (1199093)120597119891
1205971199103
120597119871
12059711990921198891199103)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119910312059711990911198891199103
+ 2119890minus2119891 sin (1199093)120597119891
1205971199103
120597119871
12059711990911198891199103
minus 119890minus2119891 cos (1199093)
1205972119871
120597119910312059711990921198891199103
+ 2119890minus2119891 cos (1199093)120597119891
1205971199103
120597119871
12059711990921198891199103)+119884
3(minus
1205972119871
120597119910312059711990931198891199103
+ 2119890minus2119891120597119891
1205971199103
120597
12059711990931198891199103)minus
120597119871
12059711991031198891199103
(18)
Using (11) we get first equations as follows
1198831[minus119890
2119891 cos (1199093)1205972119871
120597119909112059711991011198891199091
minus 11989021198912
120597119891
1205971199091cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119909112059711991021198891199091
minus 11989021198912
120597119891
1205971199091sin (1199093)
120597119871
12059711991021198891199091]
+1198832[minus119890
2119891 cos (1199093)1205972119871
120597119909212059711991011198891199091
minus 11989021198912
120597119891
1205971199092cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119909212059711991021198891199091
minus 11989021198912
120597119891
1205971199092sin (1199093)
120597119871
12059711991021198891199091]
+1198833[minus119890
2119891 cos (1199093)1205972119871
120597119909312059711991011198891199091
minus 11989021198912
120597119891
1205971199093cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119909312059711991021198891199091
minus 11989021198912
120597119891
1205971199093sin (1199093)
120597119871
12059711991021198891199091]
+1198841[minus119890
2119891 cos (1199093)1205972119871
120597119910211198891199091
minus 11989021198912
120597119891
1205971199101cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119910112059711991021198891199091
minus 11989021198912
120597119891
1205971199101sin (1199093)
120597119871
12059711991021198891199091]
+1198842[minus119890
2119891 cos (1199093)1205972119871
120597119910212059711991011198891199091
minus 11989021198912
120597119891
1205971199102cos (1199093)
120597119871
12059711991011198891199091 minus 119890
2119891 sin (1199093)1205972119871
120597119910221198891199091
minus 11989021198912
120597119891
1205971199102sin (1199093)
120597119871
12059711991021198891199091]
+1198843[minus119890
2119891 cos (1199093)1205972119871
120597119910312059711991011198891199091
minus 11989021198912
120597119891
1205971199103cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119910312059711991021198891199091
minus 11989021198912
120597119891
1205971199103sin (1199093)
120597119871
12059711991021198891199091] = minus
120597119871
12059711990911198891199091
(19)
From here
minus cos (1199093) 119881(1198902119891 120597119871
1205971199101)minus sin (1199093) 119881(119890
2119891 120597119871
1205971199102)+
120597119871
1205971199091
= 0
(20)
Advances in Mathematical Physics 9
If we think of the curve 120572 for all equations as an integralcurve of 119881 that is 119881(120572) = (120597120597119905)(120572) we find the followingequations
(PDE1) minus cos (1199093)120597
120597119905(119890
2119891 120597119871
1205971199101)
minus sin (1199093)120597
120597119905(119890
2119891 120597119871
1205971199102)+
120597119871
1205971199091= 0
(PDE2) sin (1199093)120597
120597119905(119890
2119891 120597119871
1205971199101)
minus cos (1199093)120597
120597119905(119890
2119891 120597119871
1205971199102)+
120597119871
1205971199092= 0
(PDE3) minus120597
120597119905(119890
2119891 120597119871
1205971199103)+
120597119871
1205971199093= 0
(PDE4) cos (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199091)
minus sin (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199092)+
120597119871
1205971199101= 0
(PDE5) sin (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199091)
+ cos (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199092)+
120597119871
1205971199102= 0
(PDE6) 120597
120597119905(119890minus2119891 120597119871
1205971199093)+
120597119871
1205971199103= 0
(21)
such that the differential equations (21) are named conformalEuler-Lagrange equations on conformally flat manifold whichis shown in the form of (119879119872 119892 nabla 119869) Also therefore the triple(119879119872Φ
119871 119881) is called a conformal-Lagrangian mechanical
system on (119879119872 119892 nabla 119869)
7 Weyl-Euler-Lagrangian Equations forConservative Dynamical Systems
Proposition 26 We choose 119865 = i119881 119892 = Φ
119871 and 120582 = 2119891
at (11) and by considering (4) we can write Weyl-Lagrangiandynamic equation as follows
i119881(1198902119891
Φ119871) = i119881(Φ119871) minus 119889 (2119891) (22)
The second part (11) according to the law of conservation ofenergy [32] will not change for conservative dynamical systemsand i119881(Φ119871) = Φ
119871(119881)
Φ119871(119881) minus 2119889119891 = 119889119864
119871
Φ119871(119881) = 119889119864
119871+ 2119889119891 = 119889 (119864
119871+ 2119891)
(23)
From (21) above 119871 rarr 119871 + 2119891 So we can write
(PDE7) minus cos (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199101
)
minus sin (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199102
)
+120597 (119871 + 2119891)
1205971199091= 0
(PDE8) sin (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199101
)
minus cos (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199102
)
+120597 (119871 + 2119891)
1205971199092= 0
(PDE9) minus120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199103
)+120597 (119871 + 2119891)
1205971199093= 0
(PDE10) cos (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199091)
minus sin (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199092)
+120597 (119871 + 2119891)
1205971199101= 0
(PDE11) sin (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199091)
+ cos (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199092)
+120597 (119871 + 2119891)
1205971199102= 0
(PDE12) 120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199093)+
120597 (119871 + 2119891)1205971199103
= 0
(24)
and these differential equations (24) are named Weyl-Euler-Lagrange equations for conservative dynamical systems whichare constructed on conformally flat manifold (119879119872 119892 nabla 119869 119865)
and therefore the triple (119879119872Φ119871 119881) is called a Weyl-
Lagrangian mechanical system
8 Equations Solving with Computer
Theequations systems (21) and (24) have been solved by usingthe symbolic Algebra software and implicit solution is below
119871 (1199091 1199092 1199093 1199101 1199102 1199103 119905)
= exp (minus119894 lowast 119905) lowast 1198651 (1199103 minus 119894 lowast 1199093) + 1198652 (119905)
+ exp (119905 lowast 119894) lowast 1198653 (1199103 +1199093 lowast 119894) for 119891 = 0
(25)
10 Advances in Mathematical Physics
05
1
05 1 15 2minus1
minus1
minus05
minus05
(a)
05
1
05 1 15 2minus1
minus1
minus05
minus05
(b)
Figure 1
It is well known that an electromagnetic field is a physical fieldproduced by electrically charged objects The movement ofobjects in electrical magnetic and gravitational fields force isvery important For instance on a weather map the surfacewind velocity is defined by assigning a vector to each pointon a map So each vector represents the speed and directionof the movement of air at that point
The location of each object in space is represented bythree dimensions in physical space The dimensions whichare represented by higher dimensions are time positionmass and so forth The number of dimensions of (25) will bereduced to three and behind the graphics will be drawn Firstimplicit function at (25) will be selected as special After thefigure of (25) has been drawn for the route of the movementof objects in the electromagnetic field
Example 27 Consider
119871 (1199091 1199092 1199093 1199101 1199102 1199103 119905) = exp (minus119894 lowast 119905) + exp (119905 lowast 119894) lowast 119905 minus 1199052 (26)
(see Figure 1)
9 Discussion
A classical field theory explains the study of how one or morephysical fields interact with matter which is used in quantumand classical mechanics of physics branches In this study theEuler-Lagrange mechanical equations (21) and (24) derivedon a generalized on flat manifolds may be suggested to dealwith problems in electrical magnetic and gravitational fieldsforce for the path of movement (26) of defined space movingobjects [24]
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the agency BAP of PamukkaleUniversity
References
[1] W K Liu and S Jun Computational Nanomechanics of Mate-rials American Scientific Publishers Stevenson Ranch CalifUSA 2005
[2] M Tekkoyun and Y Yayli ldquoMechanical systems on generalized-quaternionic Kahler manifoldsrdquo International Journal of Geo-metric Methods in Modern Physics vol 8 no 7 pp 1419ndash14312011
[3] D B Fogel Epistemology of a theory of everything Weyl Ein-stein and the unification of physics [PhD thesis] GraduateSchool of theUniversity ofNotreDameNotreDame Ind USA2008
[4] Z Kasap and M Tekkoyun ldquoMechanical systems on almostparapseudo-KahlerndashWeyl manifoldsrdquo International Journal ofGeometric Methods in Modern Physics vol 10 no 5 Article ID1350008 8 pages 2013
[5] Z Kasap ldquoWeyl-mechanical systems on tangent manifoldsof constant 119882-sectional curvaturerdquo International Journal ofGeometric Methods in Modern Physics vol 10 no 10 Article ID1350053 pp 1ndash13 2013
[6] M Kapovich ldquoFlat conformal structures on 3-manifolds I uni-formization of closed seifert manifoldsrdquo Journal of DifferentialGeometry vol 38 no 1 pp 191ndash215 1993
[7] F Schwartz ldquoA volumetric Penrose inequality for conformallyflat manifoldsrdquo Annales Henri Poincare vol 12 no 1 pp 67ndash762011
[8] R S Kulkarni ldquoConformally flat manifoldsrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 69 pp 2675ndash2676 1972
[9] I G Dotti and R J Miatello ldquoOn the cohomology ring offlat manifolds with a special structurerdquo Revista De La Uni OnMatematica Argentina vol 46 no 2 pp 133ndash147 2005
Advances in Mathematical Physics 11
[10] A Szczepanski Kahler at Manifolds of Low Dimensions InstitutdesHautes Etudes Scientifiques Bures-sur-Yvette France 2005
[11] R Bartnik ldquoThemass of an asymptotically flat manifoldrdquo Com-munications on Pure and AppliedMathematics vol 39 no 5 pp661ndash693 1986
[12] M D M Gonzalez ldquoSingular sets of a class of locally confor-mally flat manifoldsrdquo Duke Mathematical Journal vol 129 no3 pp 551ndash572 2005
[13] S Akbulut and M Kalafat ldquoA class of locally conformally flat4-manifoldsrdquoNewYork Journal of Mathematics vol 18 pp 733ndash763 2012
[14] S-H Zhu ldquoThe classification of complete locally conformallyflat manifolds of nonnegative Ricci curvaturerdquo Pacific Journalof Mathematics vol 163 no 1 pp 189ndash199 1994
[15] H M Abood ldquoAlmost Hermitian manifold with flat Bochnertensorrdquo European Journal of Pure and Applied Mathematics vol3 no 4 pp 730ndash736 2010
[16] K Olszak and Z Olszak ldquoOn 4-dimensional conformally flatalmost 120576-Kahlerianmanifoldsrdquo Journal of Geometry and Physicsvol 62 no 5 pp 1108ndash1113 2012
[17] S Upadhyay ldquoA bounding question for almost flat manifoldsrdquoTransactions of the AmericanMathematical Society vol 353 no3 pp 963ndash972 2001
[18] 2015 httpmathworldwolframcomFlatManifoldhtml[19] 2015 httpenwikipediaorgwikiConformally flat manifold[20] D McDu and D Salamon J-Holomorphic Curves and Quantum
Cohomology AMS 1995[21] 2015 httpenwikipediaorgwikiConformal class[22] G B Folland ldquoWeyl manifoldsrdquo Journal of Differential Geome-
try vol 4 pp 145ndash153 1970[23] L Kadosh Topics in weyl geometry [PhD thesis] University of
California Berkeley Calif USA 1996[24] H Weyl Space-Time-Matter Dover Publications 1922 Trans-
lated from the 4th German edition by H Brose MethuenLondon UK Dover Publications New York NY USA 1952
[25] P Gilkey and S Nikcevic ldquoKahler and para-Kahler curvatureWeyl manifoldsrdquo httparxivorgabs10114844
[26] H Pedersen Y S Poon and A Swann ldquoThe Einstein-Weylequations in complex and quaternionic geometryrdquo DifferentialGeometry and Its Applications vol 3 no 4 pp 309ndash321 1993
[27] P Gilkey and S Nikcevic ldquoKahler-Weylmanifolds of dimension4rdquo httparxivorgabs11094532
[28] M Brozos-Vazquez P Gilkey and E Merino ldquoGeometricrealizations of Kaehler and of para-Kaehler curvature modelsrdquoInternational Journal of Geometric Methods in Modern Physicsvol 7 no 3 pp 505ndash515 2010
[29] 2015 httpenwikipediaorgwikiWeyl tensor[30] R Miron D Hrimiuc H Shimada and S V SabauThe Geom-
etry of Hamilton and Lagrange Spaces Kluwer Academic Pub-lishers 2002
[31] J Klein ldquoEspaces variationnels et mecaniquerdquo Annales delrsquoInstitut Fourier vol 12 pp 1ndash124 1962
[32] M de Leon and P R RodriguesMethods of Differential Geom-etry in Analytical Mechanics North-Holland Elsevier Amster-dam The Netherlands 1989
[33] R Abraham J E Marsden and T Ratiu Manifolds TensorAnalysis and Applications Springer New York NY USA 2001
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
Volume 2014
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Journal of
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Advances in Mathematical Physics
R defined on every point 119901 isin 119872119872 is called a Riemannianmanifold
Definition 2 A manifold with a Riemannian metric is a flatmanifold such that it has zero curvature
Definition 3 A differentiable manifold 119872 is said to be analmost complex manifold if there exists a linear map 119869
119879119872 rarr 119879119872 satisfying 1198692 = minus119894119889 and 119869 is said to be an almostcomplex structure of119872 where 119894 is the identity (unit) operatoron 119881 such that 119881 is the vector space and 1198692 = 119869 ∘ 119869
Theorem 4 The integrability of the almost complex structureimplies a relation in the curvature Let 1199091 1199101 1199092 1199102 1199093 1199103 becoordinates on R6 with the standard flat metric
1198891199042=
3sum
119894=1(119889119909
2119894+119889119910
2119894) (1)
(see [18])
Definition 5 A (pseudo-)Riemannian manifold is confor-mally flat manifold if each point has a neighborhood that canbe mapped to flat space by a conformal transformation Let(119872 119892) be a pseudo-Riemannian manifold
Theorem 6 Let (119872 119892) be conformally flat if for each point 119909in119872 there exists a neighborhood119880 of 119909 and a smooth function119891 defined on 119880 such that (119880 1198902119891119892) is flat The function 119891 neednot be defined on all of119872 Some authors use locally conformallyflat to describe the above notion and reserve conformally flat forthe case in which the function 119891 is defined on all of119872 [19]
Definition 7 A pseudo-119869-holomorphic curve is a smoothmap from a Riemannian surface into an almost complexmanifold such that it satisfies the Cauchy-Riemann equation[20]
Definition 8 A conformal map is a function which preservesangles as the most common case where the function isbetween domains in the complex plane Conformal maps canbe defined betweendomains in higher dimensional Euclideanspaces andmore generally on a (semi-)Riemannianmanifold
Definition 9 Conformal geometry is the study of the setof angle-preserving (conformal) transformations on a spaceIn two real dimensions conformal geometry is preciselythe geometry of Riemannian surfaces In more than twodimensions conformal geometry may refer either to thestudy of conformal transformations of flat spaces (such asEuclidean spaces or spheres) or to the study of conformalmanifolds which are Riemannian or pseudo-Riemannianmanifolds with a class of metrics defined up to scale
Definition 10 A conformal manifold is a differentiable mani-fold equippedwith an equivalence class of (pseudo-)Riemannmetric tensors in which two metrics 1198921015840 and 119892 are equivalentif and only if
1198921015840
= Ψ2119892 (2)
where Ψ gt 0 is a smooth positive function An equivalenceclass of such metrics is known as a conformal metric orconformal class and a manifold with a conformal structureis called a conformal manifold [21]
3 Weyl Geometry
Conformal transformation for use in curved lengths has beenrevealed The linear distance between two points can befound easily by Riemann metric Many scientists have usedthe Riemann metric Einstein was one of the first to studythis field Einstein discovered the Riemannian geometry andsuccessfully used it to describe general relativity in the 1910that is actually a classical theory for gravitation But theuniverse is really completely not like Riemannian geometryEach path between two points is not always linear Alsoorbits of moving objects may change during movement Soeach two points in space may not be linear geodesic Thena method is required for converting nonlinear distance tolinear distance Weyl introduced a metric with a conformaltransformation in 1918The basic concepts related to the topicare listed below [22ndash24]
Definition 11 Two Riemann metrics 1198921 and 1198922 on119872 are saidto be conformally equivalent iff there exists a smooth function119891 119872 rarr R with
119890119891
1198921 = 1198922 (3)
In this case 1198921 sim 1198922
Definition 12 Let119872 be an 119899-dimensional smooth manifoldA pair (119872119866) where a conformal structure on 119872 is anequivalence class 119866 of Riemann metrics on 119872 is called aconformal structure
Theorem 13 Let nabla be a connection on119872 and 119892 isin 119866 a fixedmetric nabla is compatible with (119872119866) hArr there exists a 1-form 120596
with nabla119883119892 + 120596(119883)119892 = 0
Definition 14 A compatible torsion-free connection is calleda Weyl connection The triple (119872119866nabla) is a Weyl structure
Theorem 15 To each metric 119892 isin 119866 and 1-form 120596 there corre-sponds a unique Weyl connection nabla satisfying nabla
119883119892 +120596(119883)119892 =
0
Definition 16 Define a function 119865 1-forms on 119872 times 119866 rarr
Weyl connections by 119865(119892 120596) = nabla where nabla is the connec-tion guaranteed by Theorem 6 One says that nabla correspondsto (119892 120596)
Proposition 17 (1) 119865 is surjective
Proof 119865 is surjective byTheorem 13
(2) 119865(119892 120596) = 119865(119890119891119892 120578) iff 120578 = 120596 minus 119889119891 So
119865 (119890119891
119892) = 119865 (119892) minus 119889119891 (4)
where 119866 is a conformal structure Note that a Riemann metric119892 and a one-form 120596 determine a Weyl structure namely 119865
Advances in Mathematical Physics 3
119866 rarr and1119872 where119866 is the equivalence class of 119892 and119865(119890119891119892) =
120596 minus 119889119891
Proof Suppose that 119865(119892 120596) = 119865(119890119891119892 120578) = nabla We have
nabla119883(119890119891
119892) + 120578 (119883) 119890119891
119892
= 119883(119890119891
) 119892 + 119890119891
nabla119883119892+ 120578 (119883) 119890
119891
119892
= 119889119891 (119883) 119890119891
119892+ 119890119891
nabla119883119892+ 120578 (119883) 119890
119891
119892 = 0
(5)
Thereforenabla119883119892 = minus(119889119891(119883)+120578(119883)) On the other handnabla
119883119892+
120596(119883)119892 = 0Therefore 120596 = 120578 + 119889119891 Set nabla = 119865(119892 120596) To shownabla = 119865(119890
119891
119892 120578) and nabla119883(119890119891
119892) + 120578(119883)119890119891
119892 = 0 To calculate
nabla119883(119890119891
119892) + 120578 (119883) 119890119891
119892
= 119890119891
119889119891 (119883) 119892 + 119890119891
nabla119883119892+ (120596 (119883) minus 119889119891 (119883)) 119890
119891
119892
= 119890119891
(nabla119883119892+120596 (119883) 119892) = 0
(6)
Theorem 18 A connection on the metric bundle 120596 of aconformalmanifold119872naturally induces amap119865 119866 rarr and
1119872
and (4) and conversely Parallel translation of points in120596 by theconnection is the same as their translation by 119865
Theorem 19 Let nabla be a torsion-free connection on the tangentbundle of 119872 and 119898 ge 6 If (119872 119892 nabla 119869) is a Kahler-Weylstructure then the associated Weyl structure is trivial that isthere is a conformally equivalent metric 119892 = 119890
2119891119892 so that
(119872 119892 119869) is Kahler and so that nabla = nabla119892 [25ndash27]
Definition 20 Weyl curvature tensor is a measure of thecurvature of spacetime or a pseudo-Riemannian manifoldLike the Riemannian curvature tensor the Weyl tensorexpresses the tidal force that a body feels when moving alonga geodesic
Definition 21 Weyl transformation is a local rescaling ofthe metric tensor 119892
119886119887(119909) rarr 119890
minus2120596(119909)
119892119886119887(119909) which produces
another metric in the same conformal class A theory oran expression invariant under this transformation is calledconformally invariant or is said to possess Weyl symmetryTheWeyl symmetry is an important symmetry in conformalfield theory
4 Complex Structures on ConformallyFlat Manifold
In this sectionWeyl structures on flatmanifoldswill be trans-ferred to the mechanical system Thus the time-dependentEuler-Lagrange partial equations of motion of the dynamicsystemwill be found A flatmanifold is something that locallylooks like Euclidean space in terms of distances and anglesThe basic example is Euclidean space with the usual metric119889119904
2= sum119894119889119909
2119894 Any point on a flat manifold has a neighbor-
hood isometric to a neighborhood in Euclidean space A flatmanifold is locally Euclidean in terms of distances and anglesand merely topologically locally Euclidean as all manifolds
are The simplest nontrivial examples occur as surfaces infour-dimensional space as the flat torus is a flat manifold Itis the image of 119891(119909 119910) = (cos119909 sin119909 cos119910 sin119910)
Example 22 It vanishes if and only if 119869 is an integrable almostcomplex structure that is given any point 119875 isin 119872 there existlocal coordinates (119909
119894 119910119894) 119894 = 1 2 3 centered at 119875 following
structures taken from
1198691205971199091 = cos (1199093) 1205971199101 + sin (1199093) 1205971199102
1198691205971199092 = minus sin (1199093) 1205971199101 + cos (1199093) 1205971199102
1198691205971199093 = 1205971199103
1198691205971199101 = minus cos (1199093) 1205971199091 + sin (1199093) 1205971199092
1198691205971199102 = minus sin (1199093) 1205971199091 minus cos (1199093) 1205971199092
1198691205971199103 = minus 1205971199093
(7)
The above structures (7) have been taken from [28] We willuse 120597119909
119894= 120597120597119909
119894and 120597119910
119894= 120597120597119910
119894
The Weyl tensor differs from the Riemannian curvaturetensor in that it does not convey information on how thevolume of the body changes In dimensions 2 and 3 theWeyl curvature tensor vanishes identically Also the Weylcurvature is generally nonzero for dimensions ge4 If theWeyltensor vanishes in dimension ge4 then the metric is locallyconformally flat there exists a local coordinate system inwhich the metric tensor is proportional to a constant tensorThis fact was a key component for gravitation and generalrelativity [29]
Proposition 23 If we extend (7) by means of conformalstructure [19 30] Theorem 19 and Definition 21 we can giveequations as follows
119869120597
1205971199091= 119890
2119891 cos (1199093)120597
1205971199101+ 119890
2119891 sin (1199093)120597
1205971199102
119869120597
1205971199092= minus 119890
2119891 sin (1199093)120597
1205971199101+ 119890
2119891 cos (1199093)120597
1205971199102
119869120597
1205971199093= 119890
2119891 120597
1205971199103
119869120597
1205971199101= minus 119890minus2119891 cos (1199093)
120597
1205971199091+ 119890minus2119891 sin (1199093)
120597
1205971199092
119869120597
1205971199102= minus 119890minus2119891 sin (1199093)
120597
1205971199091minus 119890minus2119891 cos (1199093)
120597
1205971199092
119869120597
1205971199103= minus 119890minus2119891 120597
1205971199093
(8)
such that they are base structures for Weyl-Euler-Lagrangeequations where 119869 is a conformal complex structure to be simi-lar to an integrable almost complex 119869 given in (7) Fromnow onwe continue our studies thinking of the (119879119872 119892 nabla 119869) instead of
4 Advances in Mathematical Physics
Weyl manifolds (119879119872 119892 nabla 119869) Now 119869 denotes the structure ofthe holomorphic property
1198692 120597
1205971199091= 119869 ∘ 119869
120597
1205971199091= 119890
2119891 cos (1199093) 119869120597
1205971199101+ 119890
2119891 sin (1199093) 119869
sdot120597
1205971199102= 119890
2119891 cos (1199093)
sdot [minus119890minus2119891 cos (1199093)
120597
1205971199091+ 119890minus2119891 sin (1199093)
120597
1205971199092]+ 119890
2119891
sdot sin (1199093) [minus119890minus2119891 sin (1199093)
120597
1205971199091minus 119890minus2119891 cos (1199093)
120597
1205971199092]
= minus cos2 (1199093)120597
1205971199091+ cos (1199093) sin (1199093)
120597
1205971199092
minus sin2 (1199093)120597
1205971199091minus sin (1199093) cos (1199093)
120597
1205971199092
= minus [cos2 (1199093) + sin2(1199093)]
120597
1205971199091= minus
120597
1205971199091
(9)
and in similar manner it is shown that
1198692 120597
120597119909119894
= minus120597
120597119909119894
1198692 120597
120597119910119894
= minus120597
120597119910119894
119894 = 1 2 3
(10)
As can be seen from (9) and (10) 1198692 = minus119868 are the complexstructures
5 Euler-Lagrange Dynamics Equations
Definition 24 (see [31ndash33]) Let119872 be an 119899-dimensional man-ifold and 119879119872 its tangent bundle with canonical projection120591119872 119879119872 rarr 119872 119879119872 is called the phase space of velocities of
the base manifold119872 Let 119871 119879119872 rarr R be a differentiablefunction on 119879119872 and it is called the Lagrangian function Weconsider closed 2-form on 119879119872 and Φ
119871= minus119889d
119869119871 Consider
the equation
i119881Φ119871= 119889119864119871 (11)
where the semispray 119881 is a vector field Also i is a reducingfunction and i
119881Φ119871= Φ119871(119881) We will see that for motion in
a potential 119864119871= V(119871) minus 119871 is an energy function (119871 = 119879minus119875 =
(12)119898V2 minus 119898119892ℎ kinetic-potential energies) and V = 119869119881
a Liouville vector field Here 119889119864119871denotes the differential
of 119864 We will see that (11) under a certain condition on 119881is the intrinsic expression of the Euler-Lagrange equationsof motion This equation is named Euler-Lagrange dynam-ical equation The triple (119879119872Φ
119871 119881) is known as Euler-
Lagrangian systemon the tangent bundle119879119872Theoperationsrun on (11) for any coordinate system (119902
119894
(119905) 119901119894(119905)) Infinite
dimension Lagrangianrsquos equation is obtained in the formbelow
119889
119889119905(120597119871
120597 119902119894)minus
120597119871
120597119902119894= 0
119889119902119894
119889119905= 119902119894
119894 = 1 119899
(12)
6 Conformal Weyl-Euler-LagrangianEquations
Here we using (11) obtain Weyl-Euler-Lagrange equationsfor classical and quantum mechanics on conformally flatmanifold and it is shown by (119879119872 119892 nabla 119869)
Proposition 25 Let (119909119894 119910119894) be coordinate functions Also on
(119879119872 119892 nabla 119869) let 119881 be the vector field determined by 119881 =
sum3119894=1(119883119894
(120597120597119909119894) + 119884119894
(120597120597119910119894)) Then the vector field defined by
V = 119869119881
= 1198831(119890
2119891 cos (1199093)120597
1205971199101+ 119890
2119891 sin (1199093)120597
1205971199102)
+1198832(minus119890
2119891 sin (1199093)120597
1205971199101+ 119890
2119891 cos (1199093)120597
1205971199102)
+11988331198902119891 120597
1205971199103
+1198841(minus119890minus2119891 cos (1199093)
120597
1205971199091+ 119890minus2119891 sin (1199093)
120597
1205971199092)
+1198842(minus119890minus2119891 sin (1199093)
120597
1205971199091minus 119890minus2119891 cos (1199093)
120597
1205971199092)
minus1198843119890minus2119891 120597
1205971199093
(13)
is thought to be Weyl-Liouville vector field on conformally flatmanifold (119879119872 119892 nabla 119869) Φ
119871= minus119889d
119869119871 is the closed 2-form
given by (11) such that d = sum3119894=1((120597120597119909119894)119889119909119894 + (120597120597119910119894)119889119910119894)
d119869 119865(119872) rarr and
1119872 d119869= 119894119869d minus d119894
119869 and d
119869= 119869(d) =
sum3119894=1(119883119894
119869(120597120597119909119894)+119884119894
119869(120597120597119910119894)) Also the vertical differentiation
d119869is given where 119889 is the usual exterior derivationThen there
is the following result We can obtain Weyl-Euler-Lagrangeequations for classical and quantummechanics on conformallyflat manifold (119879119872 119892 nabla 119869) We get the equations given by
d119869= [119890
2119891 cos (1199093)120597
1205971199101+ 119890
2119891 sin (1199093)120597
1205971199102] 1198891199091
+[minus1198902119891 sin (1199093)
120597
1205971199101+ 119890
2119891 cos (1199093)120597
1205971199102] 1198891199092
+ 1198902119891 120597
12059711991031198891199093
Advances in Mathematical Physics 5
+[minus119890minus2119891 cos (1199093)
120597
1205971199091+ 119890minus2119891 sin (1199093)
120597
1205971199092] 1198891199101
+[minus119890minus2119891 sin (1199093)
120597
1205971199091minus 119890minus2119891 cos (1199093)
120597
1205971199092] 1198891199102
minus 119890minus2119891 120597
12059711990931198891199103
(14)
Also
Φ119871= minus119889d
119869119871
= minus119889([1198902119891 cos (1199093)
120597
1205971199101+ 119890
2119891 sin (1199093)120597
1205971199102] 1198891199091
+[minus1198902119891 sin (1199093)
120597
1205971199101+ 119890
2119891 cos (1199093)120597
1205971199102] 1198891199092
+ 1198902119891 120597119871
12059711991031198891199093
+[minus119890minus2119891 cos (1199093)
120597
1205971199091+ 119890minus2119891 sin (1199093)
120597
1205971199092] 1198891199101
+[minus119890minus2119891 sin (1199093)
120597
1205971199091minus 119890minus2119891 cos (1199093)
120597
1205971199092] 1198891199102
minus 119890minus2119891 120597119871
12059711990931198891199103)
(15)
and then we find
i119881Φ119871= Φ119871(119881) = Φ
119871(
3sum
119894=1(119883119894120597
120597119909119894
+119884119894120597
120597119910119894
)) (16)
Moreover the energy function of system is
119864119871= 119883
1[119890
2119891 cos (1199093)120597119871
1205971199101+ 119890
2119891 sin (1199093)120597119871
1205971199102]
+1198832[minus119890
2119891 sin (1199093)120597119871
1205971199101+ 119890
2119891 cos (1199093)120597119871
1205971199102]
+11988331198902119891 120597119871
1205971199103
+1198841[minus119890minus2119891 cos (1199093)
120597119871
1205971199091+ 119890minus2119891 sin (1199093)
120597119871
1205971199092]
+1198842[minus119890minus2119891 sin (1199093)
120597119871
1205971199091minus 119890minus2119891 cos (1199093)
120597119871
1205971199092]
minus1198843119890minus2119891 120597119871
1205971199093minus119871
(17)
and the differential of 119864119871is
119889119864119871= 119883
1(119890
2119891 cos (1199093)1205972119871
120597119909112059711991011198891199091
+ 21198902119891 cos (1199093)120597119891
1205971199091
120597119871
12059711991011198891199091
+ 1198902119891 sin (1199093)
1205972119871
120597119909112059711991021198891199091
+ 21198902119891 sin (1199093)120597119891
1205971199091
120597119871
12059711991021198891199091)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119909112059711991011198891199091
minus 21198902119891 sin (1199093)120597119891
1205971199091
120597119871
12059711991011198891199091
+ 1198902119891 cos (1199093)
1205972119871
120597119909112059711991021198891199091
+ 21198902119891 cos (1199093)120597119891
1205971199091
120597119871
12059711991021198891199091)
+1198833(119890
2119891 1205972119871
120597119909112059711991031198891199091 + 2119890
2119891 120597119891
1205971199091
120597119871
12059711991031198891199091)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119909211198891199091
+ 2119890minus2119891 cos (1199093)120597119891
1205971199091
120597119871
12059711990911198891199091
+ 119890minus2119891 sin (1199093)
1205972119871
120597119909112059711990921198891199091
minus 2119890minus2119891 sin (1199093)120597119891
1205971199091
120597119871
12059711990921198891199091)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119909211198891199091
+ 2119890minus2119891 sin (1199093)120597119891
1205971199091
120597119871
12059711990911198891199091
minus 119890minus2119891 cos (1199093)
1205972119871
120597119909112059711990921198891199091
+ 2119890minus2119891 cos (1199093)120597119891
1205971199091
120597119871
12059711990921198891199091)
+1198843(minus
1205972119871
120597119909112059711990931198891199091 + 2119890
minus2119891 120597119891
1205971199091
120597
12059711990931198891199091)
minus120597119871
12059711990911198891199091 +119883
1(119890
2119891 cos (1199093)1205972119871
120597119909212059711991011198891199092
6 Advances in Mathematical Physics
+ 21198902119891 cos (1199093)120597119891
1205971199092
120597119871
12059711991011198891199092
+ 1198902119891 sin (1199093)
1205972119871
120597119909212059711991021198891199092
+ 21198902119891 sin (1199093)120597119891
1205971199092
120597119871
12059711991021198891199092)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119909212059711991011198891199092
minus 21198902119891 sin (1199093)120597119891
1205971199092
120597119871
12059711991011198891199092
+ 1198902119891 cos (1199093)
1205972119871
120597119909212059711991021198891199092
+ 21198902119891 cos (1199093)120597119891
1205971199092
120597119871
12059711991021198891199092)
+1198833(119890
2119891 1205972119871
120597119909212059711991031198891199092 + 2119890
2119891 120597119891
1205971199092
120597119871
12059711991031198891199092)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119909212059711990911198891199092
+ 2119890minus2119891 cos (1199093)120597119891
1205971199092
120597119871
12059711990911198891199092
+ 119890minus2119891 sin (1199093)
1205972119871
120597119909221198891199092
minus 2119890minus2119891 sin (1199093)120597119891
1205971199092
120597119871
12059711990921198891199092)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119909212059711990911198891199092
+ 2119890minus2119891 sin (1199093)120597119891
1205971199092
120597119871
12059711990911198891199092
minus 119890minus2119891 cos (1199093)
1205972119871
120597119909221198891199092
+ 2119890minus2119891 cos (1199093)120597119891
1205971199092
120597119871
12059711990921198891199092)
+1198843(minus
1205972119871
120597119909212059711990931198891199092 + 2119890
minus2119891 120597119891
1205971199092
120597
12059711990931198891199092)
minus120597119871
12059711990921198891199092 +119883
1(119890
2119891 cos (1199093)1205972119871
120597119909312059711991011198891199093
+ 21198902119891 cos (1199093)120597119891
1205971199093
120597119871
12059711991011198891199093 minus 119890
2119891 sin (1199093)120597119871
12059711991011198891199093
+ 1198902119891 sin (1199093)
1205972119871
120597119909312059711991021198891199093
+ 21198902119891 sin (1199093)120597119891
1205971199093
120597119871
12059711991021198891199093
+ 1198902119891 cos (1199093)
120597119871
12059711991021198891199093)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119909312059711991011198891199093
minus 21198902119891 sin (1199093)120597119891
1205971199093
120597119871
12059711991011198891199093 minus 119890
2119891 cos (1199093)120597119871
12059711991011198891199093
+ 1198902119891 cos (1199093)
1205972119871
120597119909312059711991021198891199093
+ 21198902119891 cos (1199093)120597119891
1205971199093
120597119871
12059711991021198891199093
minus 1198902119891 sin (1199093)
120597119871
12059711991021198891199093)+119883
3(119890
2119891 1205972119871
120597119909312059711991031198891199093
+ 21198902119891120597119891
1205971199093
120597119871
12059711991031198891199093)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119909312059711990911198891199093
+ 2119890minus2119891 cos (1199093)120597119891
1205971199093
120597119871
12059711990911198891199093
minus 119890minus2119891 sin (1199093)
120597119871
12059711990911198891199093 + 119890
minus2119891 sin (1199093)1205972119871
120597119909312059711990921198891199093
minus 2119890minus2119891 sin (1199093)120597119891
1205971199093
120597119871
12059711990921198891199093
+ 119890minus2119891 cos (1199093)
120597119871
12059711990921198891199093)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119909312059711990911198891199093
+ 2119890minus2119891 sin (1199093)120597119891
1205971199093
120597119871
12059711990911198891199093
minus 119890minus2119891 cos (1199093)
120597119871
12059711990911198891199093
minus 119890minus2119891 cos (1199093)
1205972119871
120597119909312059711990921198891199093
+ 2119890minus2119891 cos (1199093)120597119891
1205971199093
120597119871
12059711990921198891199093
+ 119890minus2119891 sin (1199093)
120597119871
12059711990921198891199093)+119884
3(minus
1205972119871
120597119909231198891199093
+ 2119890minus2119891120597119891
1205971199093
120597
12059711990931198891199093)minus
120597119871
12059711990931198891199093
Advances in Mathematical Physics 7
+1198831(119890
2119891 cos (1199093)1205972119871
120597119910211198891199101
+ 21198902119891 cos (1199093)120597119891
1205971199101
120597119871
12059711991011198891199101
+ 1198902119891 sin (1199093)
1205972119871
120597119910112059711991021198891199101
+ 21198902119891 sin (1199093)120597119891
1205971199101
120597119871
12059711991021198891199101)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119910211198891199101
minus 21198902119891 sin (1199093)120597119891
1205971199101
120597119871
12059711991011198891199101
+ 1198902119891 cos (1199093)
1205972119871
120597119910112059711991021198891199101
+ 21198902119891 cos (1199093)120597119891
1199101
120597119871
12059711991021198891199101)+119883
3(119890
2119891 1205972119871
120597119910112059711991031198891199101
+ 21198902119891120597119891
1205971199101
120597119871
12059711991031198891199101)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119910112059711990911198891199101
+ 2119890minus2119891 cos (1199093)120597119891
1205971199101
120597119871
12059711990911198891199101
+ 119890minus2119891 sin (1199093)
1205972119871
120597119910112059711990921198891199101
minus 2119890minus2119891 sin (1199093)120597119891
1205971199101
120597119871
12059711990921198891199101)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119910112059711990911198891199101
+ 2119890minus2119891 sin (1199093)120597119891
1205971199101
120597119871
12059711990911198891199101
minus 119890minus2119891 cos (1199093)
1205972119871
120597119910112059711990921198891199101
+ 2119890minus2119891 cos (1199093)120597119891
1205971199101
120597119871
12059711990921198891199101)+119884
3(minus
1205972119871
120597119910112059711990931198891199101
+ 2119890minus2119891120597119891
1205971199101
120597
12059711990931198891199101)minus
120597119871
12059711991011198891199101
+1198831(119890
2119891 cos (1199093)1205972119871
120597119910212059711991011198891199102
+ 21198902119891 cos (1199093)120597119891
1205971199102
120597119871
12059711991011198891199102 + 119890
2119891 sin (1199093)1205972119871
120597119910221198891199102
+ 21198902119891 sin (1199093)120597119891
1205971199102
120597119871
12059711991021198891199102)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119910212059711991011198891199102
minus 21198902119891 sin (1199093)120597119891
1205971199102
120597119871
12059711991011198891199102 + 119890
2119891 cos (1199093)1205972119871
120597119910221198891199102
+ 21198902119891 cos (1199093)120597119891
1205971199102
120597119871
12059711991021198891199102)
+1198833(119890
2119891 1205972119871
120597119910212059711991031198891199102 + 2119890
2119891 120597119891
1205971199102
120597119871
12059711991031198891199102)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119910212059711990911198891199102
+ 2119890minus2119891 cos (1199093)120597119891
1205971199102
120597119871
12059711990911198891199102
+ 119890minus2119891 sin (1199093)
1205972119871
120597119910212059711990921198891199102
minus 2119890minus2119891 sin (1199093)120597119891
1205971199102
120597119871
12059711990921198891199102)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119910212059711990911198891199102
+ 2119890minus2119891 sin (1199093)120597119891
1205971199102
120597119871
12059711990911198891199102
minus 119890minus2119891 cos (1199093)
1205972119871
120597119910212059711990921198891199102
+ 2119890minus2119891 cos (1199093)120597119891
1205971199102
120597119871
12059711990921198891199102)+119884
3(minus
1205972119871
120597119910212059711990931198891199102
+ 2119890minus2119891120597119891
1205971199102
120597
12059711990931198891199102)minus
120597119871
12059711991021198891199102
+1198831(119890
2119891 cos (1199093)1205972119871
120597119910312059711991011198891199103
+ 21198902119891 cos (1199093)120597119891
1205971199103
120597119871
12059711991011198891199103
+ 1198902119891 sin (1199093)
1205972119871
120597119910312059711991021198891199103
+ 21198902119891 sin (1199093)120597119891
1205971199103
120597119871
12059711991021198891199103)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119910312059711991011198891199103
minus 21198902119891 sin (1199093)120597119891
1205971199103
120597119871
12059711991011198891199103
8 Advances in Mathematical Physics
+ 1198902119891 cos (1199093)
1205972119871
120597119910312059711991021198891199103
+ 21198902119891 cos (1199093)120597119891
1205971199103
120597119871
12059711991021198891199103)+119883
3(119890
2119891 1205972119871
120597119910231198891199103
+ 21198902119891120597119891
1205971199103
120597119871
12059711991031198891199103)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119910312059711990911198891199103
+ 2119890minus2119891 cos (1199093)120597119891
1205971199103
120597119871
12059711990911198891199103
+ 119890minus2119891 sin (1199093)
1205972119871
120597119910312059711990921198891199103
minus 2119890minus2119891 sin (1199093)120597119891
1205971199103
120597119871
12059711990921198891199103)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119910312059711990911198891199103
+ 2119890minus2119891 sin (1199093)120597119891
1205971199103
120597119871
12059711990911198891199103
minus 119890minus2119891 cos (1199093)
1205972119871
120597119910312059711990921198891199103
+ 2119890minus2119891 cos (1199093)120597119891
1205971199103
120597119871
12059711990921198891199103)+119884
3(minus
1205972119871
120597119910312059711990931198891199103
+ 2119890minus2119891120597119891
1205971199103
120597
12059711990931198891199103)minus
120597119871
12059711991031198891199103
(18)
Using (11) we get first equations as follows
1198831[minus119890
2119891 cos (1199093)1205972119871
120597119909112059711991011198891199091
minus 11989021198912
120597119891
1205971199091cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119909112059711991021198891199091
minus 11989021198912
120597119891
1205971199091sin (1199093)
120597119871
12059711991021198891199091]
+1198832[minus119890
2119891 cos (1199093)1205972119871
120597119909212059711991011198891199091
minus 11989021198912
120597119891
1205971199092cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119909212059711991021198891199091
minus 11989021198912
120597119891
1205971199092sin (1199093)
120597119871
12059711991021198891199091]
+1198833[minus119890
2119891 cos (1199093)1205972119871
120597119909312059711991011198891199091
minus 11989021198912
120597119891
1205971199093cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119909312059711991021198891199091
minus 11989021198912
120597119891
1205971199093sin (1199093)
120597119871
12059711991021198891199091]
+1198841[minus119890
2119891 cos (1199093)1205972119871
120597119910211198891199091
minus 11989021198912
120597119891
1205971199101cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119910112059711991021198891199091
minus 11989021198912
120597119891
1205971199101sin (1199093)
120597119871
12059711991021198891199091]
+1198842[minus119890
2119891 cos (1199093)1205972119871
120597119910212059711991011198891199091
minus 11989021198912
120597119891
1205971199102cos (1199093)
120597119871
12059711991011198891199091 minus 119890
2119891 sin (1199093)1205972119871
120597119910221198891199091
minus 11989021198912
120597119891
1205971199102sin (1199093)
120597119871
12059711991021198891199091]
+1198843[minus119890
2119891 cos (1199093)1205972119871
120597119910312059711991011198891199091
minus 11989021198912
120597119891
1205971199103cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119910312059711991021198891199091
minus 11989021198912
120597119891
1205971199103sin (1199093)
120597119871
12059711991021198891199091] = minus
120597119871
12059711990911198891199091
(19)
From here
minus cos (1199093) 119881(1198902119891 120597119871
1205971199101)minus sin (1199093) 119881(119890
2119891 120597119871
1205971199102)+
120597119871
1205971199091
= 0
(20)
Advances in Mathematical Physics 9
If we think of the curve 120572 for all equations as an integralcurve of 119881 that is 119881(120572) = (120597120597119905)(120572) we find the followingequations
(PDE1) minus cos (1199093)120597
120597119905(119890
2119891 120597119871
1205971199101)
minus sin (1199093)120597
120597119905(119890
2119891 120597119871
1205971199102)+
120597119871
1205971199091= 0
(PDE2) sin (1199093)120597
120597119905(119890
2119891 120597119871
1205971199101)
minus cos (1199093)120597
120597119905(119890
2119891 120597119871
1205971199102)+
120597119871
1205971199092= 0
(PDE3) minus120597
120597119905(119890
2119891 120597119871
1205971199103)+
120597119871
1205971199093= 0
(PDE4) cos (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199091)
minus sin (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199092)+
120597119871
1205971199101= 0
(PDE5) sin (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199091)
+ cos (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199092)+
120597119871
1205971199102= 0
(PDE6) 120597
120597119905(119890minus2119891 120597119871
1205971199093)+
120597119871
1205971199103= 0
(21)
such that the differential equations (21) are named conformalEuler-Lagrange equations on conformally flat manifold whichis shown in the form of (119879119872 119892 nabla 119869) Also therefore the triple(119879119872Φ
119871 119881) is called a conformal-Lagrangian mechanical
system on (119879119872 119892 nabla 119869)
7 Weyl-Euler-Lagrangian Equations forConservative Dynamical Systems
Proposition 26 We choose 119865 = i119881 119892 = Φ
119871 and 120582 = 2119891
at (11) and by considering (4) we can write Weyl-Lagrangiandynamic equation as follows
i119881(1198902119891
Φ119871) = i119881(Φ119871) minus 119889 (2119891) (22)
The second part (11) according to the law of conservation ofenergy [32] will not change for conservative dynamical systemsand i119881(Φ119871) = Φ
119871(119881)
Φ119871(119881) minus 2119889119891 = 119889119864
119871
Φ119871(119881) = 119889119864
119871+ 2119889119891 = 119889 (119864
119871+ 2119891)
(23)
From (21) above 119871 rarr 119871 + 2119891 So we can write
(PDE7) minus cos (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199101
)
minus sin (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199102
)
+120597 (119871 + 2119891)
1205971199091= 0
(PDE8) sin (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199101
)
minus cos (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199102
)
+120597 (119871 + 2119891)
1205971199092= 0
(PDE9) minus120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199103
)+120597 (119871 + 2119891)
1205971199093= 0
(PDE10) cos (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199091)
minus sin (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199092)
+120597 (119871 + 2119891)
1205971199101= 0
(PDE11) sin (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199091)
+ cos (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199092)
+120597 (119871 + 2119891)
1205971199102= 0
(PDE12) 120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199093)+
120597 (119871 + 2119891)1205971199103
= 0
(24)
and these differential equations (24) are named Weyl-Euler-Lagrange equations for conservative dynamical systems whichare constructed on conformally flat manifold (119879119872 119892 nabla 119869 119865)
and therefore the triple (119879119872Φ119871 119881) is called a Weyl-
Lagrangian mechanical system
8 Equations Solving with Computer
Theequations systems (21) and (24) have been solved by usingthe symbolic Algebra software and implicit solution is below
119871 (1199091 1199092 1199093 1199101 1199102 1199103 119905)
= exp (minus119894 lowast 119905) lowast 1198651 (1199103 minus 119894 lowast 1199093) + 1198652 (119905)
+ exp (119905 lowast 119894) lowast 1198653 (1199103 +1199093 lowast 119894) for 119891 = 0
(25)
10 Advances in Mathematical Physics
05
1
05 1 15 2minus1
minus1
minus05
minus05
(a)
05
1
05 1 15 2minus1
minus1
minus05
minus05
(b)
Figure 1
It is well known that an electromagnetic field is a physical fieldproduced by electrically charged objects The movement ofobjects in electrical magnetic and gravitational fields force isvery important For instance on a weather map the surfacewind velocity is defined by assigning a vector to each pointon a map So each vector represents the speed and directionof the movement of air at that point
The location of each object in space is represented bythree dimensions in physical space The dimensions whichare represented by higher dimensions are time positionmass and so forth The number of dimensions of (25) will bereduced to three and behind the graphics will be drawn Firstimplicit function at (25) will be selected as special After thefigure of (25) has been drawn for the route of the movementof objects in the electromagnetic field
Example 27 Consider
119871 (1199091 1199092 1199093 1199101 1199102 1199103 119905) = exp (minus119894 lowast 119905) + exp (119905 lowast 119894) lowast 119905 minus 1199052 (26)
(see Figure 1)
9 Discussion
A classical field theory explains the study of how one or morephysical fields interact with matter which is used in quantumand classical mechanics of physics branches In this study theEuler-Lagrange mechanical equations (21) and (24) derivedon a generalized on flat manifolds may be suggested to dealwith problems in electrical magnetic and gravitational fieldsforce for the path of movement (26) of defined space movingobjects [24]
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the agency BAP of PamukkaleUniversity
References
[1] W K Liu and S Jun Computational Nanomechanics of Mate-rials American Scientific Publishers Stevenson Ranch CalifUSA 2005
[2] M Tekkoyun and Y Yayli ldquoMechanical systems on generalized-quaternionic Kahler manifoldsrdquo International Journal of Geo-metric Methods in Modern Physics vol 8 no 7 pp 1419ndash14312011
[3] D B Fogel Epistemology of a theory of everything Weyl Ein-stein and the unification of physics [PhD thesis] GraduateSchool of theUniversity ofNotreDameNotreDame Ind USA2008
[4] Z Kasap and M Tekkoyun ldquoMechanical systems on almostparapseudo-KahlerndashWeyl manifoldsrdquo International Journal ofGeometric Methods in Modern Physics vol 10 no 5 Article ID1350008 8 pages 2013
[5] Z Kasap ldquoWeyl-mechanical systems on tangent manifoldsof constant 119882-sectional curvaturerdquo International Journal ofGeometric Methods in Modern Physics vol 10 no 10 Article ID1350053 pp 1ndash13 2013
[6] M Kapovich ldquoFlat conformal structures on 3-manifolds I uni-formization of closed seifert manifoldsrdquo Journal of DifferentialGeometry vol 38 no 1 pp 191ndash215 1993
[7] F Schwartz ldquoA volumetric Penrose inequality for conformallyflat manifoldsrdquo Annales Henri Poincare vol 12 no 1 pp 67ndash762011
[8] R S Kulkarni ldquoConformally flat manifoldsrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 69 pp 2675ndash2676 1972
[9] I G Dotti and R J Miatello ldquoOn the cohomology ring offlat manifolds with a special structurerdquo Revista De La Uni OnMatematica Argentina vol 46 no 2 pp 133ndash147 2005
Advances in Mathematical Physics 11
[10] A Szczepanski Kahler at Manifolds of Low Dimensions InstitutdesHautes Etudes Scientifiques Bures-sur-Yvette France 2005
[11] R Bartnik ldquoThemass of an asymptotically flat manifoldrdquo Com-munications on Pure and AppliedMathematics vol 39 no 5 pp661ndash693 1986
[12] M D M Gonzalez ldquoSingular sets of a class of locally confor-mally flat manifoldsrdquo Duke Mathematical Journal vol 129 no3 pp 551ndash572 2005
[13] S Akbulut and M Kalafat ldquoA class of locally conformally flat4-manifoldsrdquoNewYork Journal of Mathematics vol 18 pp 733ndash763 2012
[14] S-H Zhu ldquoThe classification of complete locally conformallyflat manifolds of nonnegative Ricci curvaturerdquo Pacific Journalof Mathematics vol 163 no 1 pp 189ndash199 1994
[15] H M Abood ldquoAlmost Hermitian manifold with flat Bochnertensorrdquo European Journal of Pure and Applied Mathematics vol3 no 4 pp 730ndash736 2010
[16] K Olszak and Z Olszak ldquoOn 4-dimensional conformally flatalmost 120576-Kahlerianmanifoldsrdquo Journal of Geometry and Physicsvol 62 no 5 pp 1108ndash1113 2012
[17] S Upadhyay ldquoA bounding question for almost flat manifoldsrdquoTransactions of the AmericanMathematical Society vol 353 no3 pp 963ndash972 2001
[18] 2015 httpmathworldwolframcomFlatManifoldhtml[19] 2015 httpenwikipediaorgwikiConformally flat manifold[20] D McDu and D Salamon J-Holomorphic Curves and Quantum
Cohomology AMS 1995[21] 2015 httpenwikipediaorgwikiConformal class[22] G B Folland ldquoWeyl manifoldsrdquo Journal of Differential Geome-
try vol 4 pp 145ndash153 1970[23] L Kadosh Topics in weyl geometry [PhD thesis] University of
California Berkeley Calif USA 1996[24] H Weyl Space-Time-Matter Dover Publications 1922 Trans-
lated from the 4th German edition by H Brose MethuenLondon UK Dover Publications New York NY USA 1952
[25] P Gilkey and S Nikcevic ldquoKahler and para-Kahler curvatureWeyl manifoldsrdquo httparxivorgabs10114844
[26] H Pedersen Y S Poon and A Swann ldquoThe Einstein-Weylequations in complex and quaternionic geometryrdquo DifferentialGeometry and Its Applications vol 3 no 4 pp 309ndash321 1993
[27] P Gilkey and S Nikcevic ldquoKahler-Weylmanifolds of dimension4rdquo httparxivorgabs11094532
[28] M Brozos-Vazquez P Gilkey and E Merino ldquoGeometricrealizations of Kaehler and of para-Kaehler curvature modelsrdquoInternational Journal of Geometric Methods in Modern Physicsvol 7 no 3 pp 505ndash515 2010
[29] 2015 httpenwikipediaorgwikiWeyl tensor[30] R Miron D Hrimiuc H Shimada and S V SabauThe Geom-
etry of Hamilton and Lagrange Spaces Kluwer Academic Pub-lishers 2002
[31] J Klein ldquoEspaces variationnels et mecaniquerdquo Annales delrsquoInstitut Fourier vol 12 pp 1ndash124 1962
[32] M de Leon and P R RodriguesMethods of Differential Geom-etry in Analytical Mechanics North-Holland Elsevier Amster-dam The Netherlands 1989
[33] R Abraham J E Marsden and T Ratiu Manifolds TensorAnalysis and Applications Springer New York NY USA 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 3
119866 rarr and1119872 where119866 is the equivalence class of 119892 and119865(119890119891119892) =
120596 minus 119889119891
Proof Suppose that 119865(119892 120596) = 119865(119890119891119892 120578) = nabla We have
nabla119883(119890119891
119892) + 120578 (119883) 119890119891
119892
= 119883(119890119891
) 119892 + 119890119891
nabla119883119892+ 120578 (119883) 119890
119891
119892
= 119889119891 (119883) 119890119891
119892+ 119890119891
nabla119883119892+ 120578 (119883) 119890
119891
119892 = 0
(5)
Thereforenabla119883119892 = minus(119889119891(119883)+120578(119883)) On the other handnabla
119883119892+
120596(119883)119892 = 0Therefore 120596 = 120578 + 119889119891 Set nabla = 119865(119892 120596) To shownabla = 119865(119890
119891
119892 120578) and nabla119883(119890119891
119892) + 120578(119883)119890119891
119892 = 0 To calculate
nabla119883(119890119891
119892) + 120578 (119883) 119890119891
119892
= 119890119891
119889119891 (119883) 119892 + 119890119891
nabla119883119892+ (120596 (119883) minus 119889119891 (119883)) 119890
119891
119892
= 119890119891
(nabla119883119892+120596 (119883) 119892) = 0
(6)
Theorem 18 A connection on the metric bundle 120596 of aconformalmanifold119872naturally induces amap119865 119866 rarr and
1119872
and (4) and conversely Parallel translation of points in120596 by theconnection is the same as their translation by 119865
Theorem 19 Let nabla be a torsion-free connection on the tangentbundle of 119872 and 119898 ge 6 If (119872 119892 nabla 119869) is a Kahler-Weylstructure then the associated Weyl structure is trivial that isthere is a conformally equivalent metric 119892 = 119890
2119891119892 so that
(119872 119892 119869) is Kahler and so that nabla = nabla119892 [25ndash27]
Definition 20 Weyl curvature tensor is a measure of thecurvature of spacetime or a pseudo-Riemannian manifoldLike the Riemannian curvature tensor the Weyl tensorexpresses the tidal force that a body feels when moving alonga geodesic
Definition 21 Weyl transformation is a local rescaling ofthe metric tensor 119892
119886119887(119909) rarr 119890
minus2120596(119909)
119892119886119887(119909) which produces
another metric in the same conformal class A theory oran expression invariant under this transformation is calledconformally invariant or is said to possess Weyl symmetryTheWeyl symmetry is an important symmetry in conformalfield theory
4 Complex Structures on ConformallyFlat Manifold
In this sectionWeyl structures on flatmanifoldswill be trans-ferred to the mechanical system Thus the time-dependentEuler-Lagrange partial equations of motion of the dynamicsystemwill be found A flatmanifold is something that locallylooks like Euclidean space in terms of distances and anglesThe basic example is Euclidean space with the usual metric119889119904
2= sum119894119889119909
2119894 Any point on a flat manifold has a neighbor-
hood isometric to a neighborhood in Euclidean space A flatmanifold is locally Euclidean in terms of distances and anglesand merely topologically locally Euclidean as all manifolds
are The simplest nontrivial examples occur as surfaces infour-dimensional space as the flat torus is a flat manifold Itis the image of 119891(119909 119910) = (cos119909 sin119909 cos119910 sin119910)
Example 22 It vanishes if and only if 119869 is an integrable almostcomplex structure that is given any point 119875 isin 119872 there existlocal coordinates (119909
119894 119910119894) 119894 = 1 2 3 centered at 119875 following
structures taken from
1198691205971199091 = cos (1199093) 1205971199101 + sin (1199093) 1205971199102
1198691205971199092 = minus sin (1199093) 1205971199101 + cos (1199093) 1205971199102
1198691205971199093 = 1205971199103
1198691205971199101 = minus cos (1199093) 1205971199091 + sin (1199093) 1205971199092
1198691205971199102 = minus sin (1199093) 1205971199091 minus cos (1199093) 1205971199092
1198691205971199103 = minus 1205971199093
(7)
The above structures (7) have been taken from [28] We willuse 120597119909
119894= 120597120597119909
119894and 120597119910
119894= 120597120597119910
119894
The Weyl tensor differs from the Riemannian curvaturetensor in that it does not convey information on how thevolume of the body changes In dimensions 2 and 3 theWeyl curvature tensor vanishes identically Also the Weylcurvature is generally nonzero for dimensions ge4 If theWeyltensor vanishes in dimension ge4 then the metric is locallyconformally flat there exists a local coordinate system inwhich the metric tensor is proportional to a constant tensorThis fact was a key component for gravitation and generalrelativity [29]
Proposition 23 If we extend (7) by means of conformalstructure [19 30] Theorem 19 and Definition 21 we can giveequations as follows
119869120597
1205971199091= 119890
2119891 cos (1199093)120597
1205971199101+ 119890
2119891 sin (1199093)120597
1205971199102
119869120597
1205971199092= minus 119890
2119891 sin (1199093)120597
1205971199101+ 119890
2119891 cos (1199093)120597
1205971199102
119869120597
1205971199093= 119890
2119891 120597
1205971199103
119869120597
1205971199101= minus 119890minus2119891 cos (1199093)
120597
1205971199091+ 119890minus2119891 sin (1199093)
120597
1205971199092
119869120597
1205971199102= minus 119890minus2119891 sin (1199093)
120597
1205971199091minus 119890minus2119891 cos (1199093)
120597
1205971199092
119869120597
1205971199103= minus 119890minus2119891 120597
1205971199093
(8)
such that they are base structures for Weyl-Euler-Lagrangeequations where 119869 is a conformal complex structure to be simi-lar to an integrable almost complex 119869 given in (7) Fromnow onwe continue our studies thinking of the (119879119872 119892 nabla 119869) instead of
4 Advances in Mathematical Physics
Weyl manifolds (119879119872 119892 nabla 119869) Now 119869 denotes the structure ofthe holomorphic property
1198692 120597
1205971199091= 119869 ∘ 119869
120597
1205971199091= 119890
2119891 cos (1199093) 119869120597
1205971199101+ 119890
2119891 sin (1199093) 119869
sdot120597
1205971199102= 119890
2119891 cos (1199093)
sdot [minus119890minus2119891 cos (1199093)
120597
1205971199091+ 119890minus2119891 sin (1199093)
120597
1205971199092]+ 119890
2119891
sdot sin (1199093) [minus119890minus2119891 sin (1199093)
120597
1205971199091minus 119890minus2119891 cos (1199093)
120597
1205971199092]
= minus cos2 (1199093)120597
1205971199091+ cos (1199093) sin (1199093)
120597
1205971199092
minus sin2 (1199093)120597
1205971199091minus sin (1199093) cos (1199093)
120597
1205971199092
= minus [cos2 (1199093) + sin2(1199093)]
120597
1205971199091= minus
120597
1205971199091
(9)
and in similar manner it is shown that
1198692 120597
120597119909119894
= minus120597
120597119909119894
1198692 120597
120597119910119894
= minus120597
120597119910119894
119894 = 1 2 3
(10)
As can be seen from (9) and (10) 1198692 = minus119868 are the complexstructures
5 Euler-Lagrange Dynamics Equations
Definition 24 (see [31ndash33]) Let119872 be an 119899-dimensional man-ifold and 119879119872 its tangent bundle with canonical projection120591119872 119879119872 rarr 119872 119879119872 is called the phase space of velocities of
the base manifold119872 Let 119871 119879119872 rarr R be a differentiablefunction on 119879119872 and it is called the Lagrangian function Weconsider closed 2-form on 119879119872 and Φ
119871= minus119889d
119869119871 Consider
the equation
i119881Φ119871= 119889119864119871 (11)
where the semispray 119881 is a vector field Also i is a reducingfunction and i
119881Φ119871= Φ119871(119881) We will see that for motion in
a potential 119864119871= V(119871) minus 119871 is an energy function (119871 = 119879minus119875 =
(12)119898V2 minus 119898119892ℎ kinetic-potential energies) and V = 119869119881
a Liouville vector field Here 119889119864119871denotes the differential
of 119864 We will see that (11) under a certain condition on 119881is the intrinsic expression of the Euler-Lagrange equationsof motion This equation is named Euler-Lagrange dynam-ical equation The triple (119879119872Φ
119871 119881) is known as Euler-
Lagrangian systemon the tangent bundle119879119872Theoperationsrun on (11) for any coordinate system (119902
119894
(119905) 119901119894(119905)) Infinite
dimension Lagrangianrsquos equation is obtained in the formbelow
119889
119889119905(120597119871
120597 119902119894)minus
120597119871
120597119902119894= 0
119889119902119894
119889119905= 119902119894
119894 = 1 119899
(12)
6 Conformal Weyl-Euler-LagrangianEquations
Here we using (11) obtain Weyl-Euler-Lagrange equationsfor classical and quantum mechanics on conformally flatmanifold and it is shown by (119879119872 119892 nabla 119869)
Proposition 25 Let (119909119894 119910119894) be coordinate functions Also on
(119879119872 119892 nabla 119869) let 119881 be the vector field determined by 119881 =
sum3119894=1(119883119894
(120597120597119909119894) + 119884119894
(120597120597119910119894)) Then the vector field defined by
V = 119869119881
= 1198831(119890
2119891 cos (1199093)120597
1205971199101+ 119890
2119891 sin (1199093)120597
1205971199102)
+1198832(minus119890
2119891 sin (1199093)120597
1205971199101+ 119890
2119891 cos (1199093)120597
1205971199102)
+11988331198902119891 120597
1205971199103
+1198841(minus119890minus2119891 cos (1199093)
120597
1205971199091+ 119890minus2119891 sin (1199093)
120597
1205971199092)
+1198842(minus119890minus2119891 sin (1199093)
120597
1205971199091minus 119890minus2119891 cos (1199093)
120597
1205971199092)
minus1198843119890minus2119891 120597
1205971199093
(13)
is thought to be Weyl-Liouville vector field on conformally flatmanifold (119879119872 119892 nabla 119869) Φ
119871= minus119889d
119869119871 is the closed 2-form
given by (11) such that d = sum3119894=1((120597120597119909119894)119889119909119894 + (120597120597119910119894)119889119910119894)
d119869 119865(119872) rarr and
1119872 d119869= 119894119869d minus d119894
119869 and d
119869= 119869(d) =
sum3119894=1(119883119894
119869(120597120597119909119894)+119884119894
119869(120597120597119910119894)) Also the vertical differentiation
d119869is given where 119889 is the usual exterior derivationThen there
is the following result We can obtain Weyl-Euler-Lagrangeequations for classical and quantummechanics on conformallyflat manifold (119879119872 119892 nabla 119869) We get the equations given by
d119869= [119890
2119891 cos (1199093)120597
1205971199101+ 119890
2119891 sin (1199093)120597
1205971199102] 1198891199091
+[minus1198902119891 sin (1199093)
120597
1205971199101+ 119890
2119891 cos (1199093)120597
1205971199102] 1198891199092
+ 1198902119891 120597
12059711991031198891199093
Advances in Mathematical Physics 5
+[minus119890minus2119891 cos (1199093)
120597
1205971199091+ 119890minus2119891 sin (1199093)
120597
1205971199092] 1198891199101
+[minus119890minus2119891 sin (1199093)
120597
1205971199091minus 119890minus2119891 cos (1199093)
120597
1205971199092] 1198891199102
minus 119890minus2119891 120597
12059711990931198891199103
(14)
Also
Φ119871= minus119889d
119869119871
= minus119889([1198902119891 cos (1199093)
120597
1205971199101+ 119890
2119891 sin (1199093)120597
1205971199102] 1198891199091
+[minus1198902119891 sin (1199093)
120597
1205971199101+ 119890
2119891 cos (1199093)120597
1205971199102] 1198891199092
+ 1198902119891 120597119871
12059711991031198891199093
+[minus119890minus2119891 cos (1199093)
120597
1205971199091+ 119890minus2119891 sin (1199093)
120597
1205971199092] 1198891199101
+[minus119890minus2119891 sin (1199093)
120597
1205971199091minus 119890minus2119891 cos (1199093)
120597
1205971199092] 1198891199102
minus 119890minus2119891 120597119871
12059711990931198891199103)
(15)
and then we find
i119881Φ119871= Φ119871(119881) = Φ
119871(
3sum
119894=1(119883119894120597
120597119909119894
+119884119894120597
120597119910119894
)) (16)
Moreover the energy function of system is
119864119871= 119883
1[119890
2119891 cos (1199093)120597119871
1205971199101+ 119890
2119891 sin (1199093)120597119871
1205971199102]
+1198832[minus119890
2119891 sin (1199093)120597119871
1205971199101+ 119890
2119891 cos (1199093)120597119871
1205971199102]
+11988331198902119891 120597119871
1205971199103
+1198841[minus119890minus2119891 cos (1199093)
120597119871
1205971199091+ 119890minus2119891 sin (1199093)
120597119871
1205971199092]
+1198842[minus119890minus2119891 sin (1199093)
120597119871
1205971199091minus 119890minus2119891 cos (1199093)
120597119871
1205971199092]
minus1198843119890minus2119891 120597119871
1205971199093minus119871
(17)
and the differential of 119864119871is
119889119864119871= 119883
1(119890
2119891 cos (1199093)1205972119871
120597119909112059711991011198891199091
+ 21198902119891 cos (1199093)120597119891
1205971199091
120597119871
12059711991011198891199091
+ 1198902119891 sin (1199093)
1205972119871
120597119909112059711991021198891199091
+ 21198902119891 sin (1199093)120597119891
1205971199091
120597119871
12059711991021198891199091)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119909112059711991011198891199091
minus 21198902119891 sin (1199093)120597119891
1205971199091
120597119871
12059711991011198891199091
+ 1198902119891 cos (1199093)
1205972119871
120597119909112059711991021198891199091
+ 21198902119891 cos (1199093)120597119891
1205971199091
120597119871
12059711991021198891199091)
+1198833(119890
2119891 1205972119871
120597119909112059711991031198891199091 + 2119890
2119891 120597119891
1205971199091
120597119871
12059711991031198891199091)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119909211198891199091
+ 2119890minus2119891 cos (1199093)120597119891
1205971199091
120597119871
12059711990911198891199091
+ 119890minus2119891 sin (1199093)
1205972119871
120597119909112059711990921198891199091
minus 2119890minus2119891 sin (1199093)120597119891
1205971199091
120597119871
12059711990921198891199091)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119909211198891199091
+ 2119890minus2119891 sin (1199093)120597119891
1205971199091
120597119871
12059711990911198891199091
minus 119890minus2119891 cos (1199093)
1205972119871
120597119909112059711990921198891199091
+ 2119890minus2119891 cos (1199093)120597119891
1205971199091
120597119871
12059711990921198891199091)
+1198843(minus
1205972119871
120597119909112059711990931198891199091 + 2119890
minus2119891 120597119891
1205971199091
120597
12059711990931198891199091)
minus120597119871
12059711990911198891199091 +119883
1(119890
2119891 cos (1199093)1205972119871
120597119909212059711991011198891199092
6 Advances in Mathematical Physics
+ 21198902119891 cos (1199093)120597119891
1205971199092
120597119871
12059711991011198891199092
+ 1198902119891 sin (1199093)
1205972119871
120597119909212059711991021198891199092
+ 21198902119891 sin (1199093)120597119891
1205971199092
120597119871
12059711991021198891199092)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119909212059711991011198891199092
minus 21198902119891 sin (1199093)120597119891
1205971199092
120597119871
12059711991011198891199092
+ 1198902119891 cos (1199093)
1205972119871
120597119909212059711991021198891199092
+ 21198902119891 cos (1199093)120597119891
1205971199092
120597119871
12059711991021198891199092)
+1198833(119890
2119891 1205972119871
120597119909212059711991031198891199092 + 2119890
2119891 120597119891
1205971199092
120597119871
12059711991031198891199092)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119909212059711990911198891199092
+ 2119890minus2119891 cos (1199093)120597119891
1205971199092
120597119871
12059711990911198891199092
+ 119890minus2119891 sin (1199093)
1205972119871
120597119909221198891199092
minus 2119890minus2119891 sin (1199093)120597119891
1205971199092
120597119871
12059711990921198891199092)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119909212059711990911198891199092
+ 2119890minus2119891 sin (1199093)120597119891
1205971199092
120597119871
12059711990911198891199092
minus 119890minus2119891 cos (1199093)
1205972119871
120597119909221198891199092
+ 2119890minus2119891 cos (1199093)120597119891
1205971199092
120597119871
12059711990921198891199092)
+1198843(minus
1205972119871
120597119909212059711990931198891199092 + 2119890
minus2119891 120597119891
1205971199092
120597
12059711990931198891199092)
minus120597119871
12059711990921198891199092 +119883
1(119890
2119891 cos (1199093)1205972119871
120597119909312059711991011198891199093
+ 21198902119891 cos (1199093)120597119891
1205971199093
120597119871
12059711991011198891199093 minus 119890
2119891 sin (1199093)120597119871
12059711991011198891199093
+ 1198902119891 sin (1199093)
1205972119871
120597119909312059711991021198891199093
+ 21198902119891 sin (1199093)120597119891
1205971199093
120597119871
12059711991021198891199093
+ 1198902119891 cos (1199093)
120597119871
12059711991021198891199093)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119909312059711991011198891199093
minus 21198902119891 sin (1199093)120597119891
1205971199093
120597119871
12059711991011198891199093 minus 119890
2119891 cos (1199093)120597119871
12059711991011198891199093
+ 1198902119891 cos (1199093)
1205972119871
120597119909312059711991021198891199093
+ 21198902119891 cos (1199093)120597119891
1205971199093
120597119871
12059711991021198891199093
minus 1198902119891 sin (1199093)
120597119871
12059711991021198891199093)+119883
3(119890
2119891 1205972119871
120597119909312059711991031198891199093
+ 21198902119891120597119891
1205971199093
120597119871
12059711991031198891199093)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119909312059711990911198891199093
+ 2119890minus2119891 cos (1199093)120597119891
1205971199093
120597119871
12059711990911198891199093
minus 119890minus2119891 sin (1199093)
120597119871
12059711990911198891199093 + 119890
minus2119891 sin (1199093)1205972119871
120597119909312059711990921198891199093
minus 2119890minus2119891 sin (1199093)120597119891
1205971199093
120597119871
12059711990921198891199093
+ 119890minus2119891 cos (1199093)
120597119871
12059711990921198891199093)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119909312059711990911198891199093
+ 2119890minus2119891 sin (1199093)120597119891
1205971199093
120597119871
12059711990911198891199093
minus 119890minus2119891 cos (1199093)
120597119871
12059711990911198891199093
minus 119890minus2119891 cos (1199093)
1205972119871
120597119909312059711990921198891199093
+ 2119890minus2119891 cos (1199093)120597119891
1205971199093
120597119871
12059711990921198891199093
+ 119890minus2119891 sin (1199093)
120597119871
12059711990921198891199093)+119884
3(minus
1205972119871
120597119909231198891199093
+ 2119890minus2119891120597119891
1205971199093
120597
12059711990931198891199093)minus
120597119871
12059711990931198891199093
Advances in Mathematical Physics 7
+1198831(119890
2119891 cos (1199093)1205972119871
120597119910211198891199101
+ 21198902119891 cos (1199093)120597119891
1205971199101
120597119871
12059711991011198891199101
+ 1198902119891 sin (1199093)
1205972119871
120597119910112059711991021198891199101
+ 21198902119891 sin (1199093)120597119891
1205971199101
120597119871
12059711991021198891199101)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119910211198891199101
minus 21198902119891 sin (1199093)120597119891
1205971199101
120597119871
12059711991011198891199101
+ 1198902119891 cos (1199093)
1205972119871
120597119910112059711991021198891199101
+ 21198902119891 cos (1199093)120597119891
1199101
120597119871
12059711991021198891199101)+119883
3(119890
2119891 1205972119871
120597119910112059711991031198891199101
+ 21198902119891120597119891
1205971199101
120597119871
12059711991031198891199101)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119910112059711990911198891199101
+ 2119890minus2119891 cos (1199093)120597119891
1205971199101
120597119871
12059711990911198891199101
+ 119890minus2119891 sin (1199093)
1205972119871
120597119910112059711990921198891199101
minus 2119890minus2119891 sin (1199093)120597119891
1205971199101
120597119871
12059711990921198891199101)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119910112059711990911198891199101
+ 2119890minus2119891 sin (1199093)120597119891
1205971199101
120597119871
12059711990911198891199101
minus 119890minus2119891 cos (1199093)
1205972119871
120597119910112059711990921198891199101
+ 2119890minus2119891 cos (1199093)120597119891
1205971199101
120597119871
12059711990921198891199101)+119884
3(minus
1205972119871
120597119910112059711990931198891199101
+ 2119890minus2119891120597119891
1205971199101
120597
12059711990931198891199101)minus
120597119871
12059711991011198891199101
+1198831(119890
2119891 cos (1199093)1205972119871
120597119910212059711991011198891199102
+ 21198902119891 cos (1199093)120597119891
1205971199102
120597119871
12059711991011198891199102 + 119890
2119891 sin (1199093)1205972119871
120597119910221198891199102
+ 21198902119891 sin (1199093)120597119891
1205971199102
120597119871
12059711991021198891199102)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119910212059711991011198891199102
minus 21198902119891 sin (1199093)120597119891
1205971199102
120597119871
12059711991011198891199102 + 119890
2119891 cos (1199093)1205972119871
120597119910221198891199102
+ 21198902119891 cos (1199093)120597119891
1205971199102
120597119871
12059711991021198891199102)
+1198833(119890
2119891 1205972119871
120597119910212059711991031198891199102 + 2119890
2119891 120597119891
1205971199102
120597119871
12059711991031198891199102)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119910212059711990911198891199102
+ 2119890minus2119891 cos (1199093)120597119891
1205971199102
120597119871
12059711990911198891199102
+ 119890minus2119891 sin (1199093)
1205972119871
120597119910212059711990921198891199102
minus 2119890minus2119891 sin (1199093)120597119891
1205971199102
120597119871
12059711990921198891199102)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119910212059711990911198891199102
+ 2119890minus2119891 sin (1199093)120597119891
1205971199102
120597119871
12059711990911198891199102
minus 119890minus2119891 cos (1199093)
1205972119871
120597119910212059711990921198891199102
+ 2119890minus2119891 cos (1199093)120597119891
1205971199102
120597119871
12059711990921198891199102)+119884
3(minus
1205972119871
120597119910212059711990931198891199102
+ 2119890minus2119891120597119891
1205971199102
120597
12059711990931198891199102)minus
120597119871
12059711991021198891199102
+1198831(119890
2119891 cos (1199093)1205972119871
120597119910312059711991011198891199103
+ 21198902119891 cos (1199093)120597119891
1205971199103
120597119871
12059711991011198891199103
+ 1198902119891 sin (1199093)
1205972119871
120597119910312059711991021198891199103
+ 21198902119891 sin (1199093)120597119891
1205971199103
120597119871
12059711991021198891199103)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119910312059711991011198891199103
minus 21198902119891 sin (1199093)120597119891
1205971199103
120597119871
12059711991011198891199103
8 Advances in Mathematical Physics
+ 1198902119891 cos (1199093)
1205972119871
120597119910312059711991021198891199103
+ 21198902119891 cos (1199093)120597119891
1205971199103
120597119871
12059711991021198891199103)+119883
3(119890
2119891 1205972119871
120597119910231198891199103
+ 21198902119891120597119891
1205971199103
120597119871
12059711991031198891199103)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119910312059711990911198891199103
+ 2119890minus2119891 cos (1199093)120597119891
1205971199103
120597119871
12059711990911198891199103
+ 119890minus2119891 sin (1199093)
1205972119871
120597119910312059711990921198891199103
minus 2119890minus2119891 sin (1199093)120597119891
1205971199103
120597119871
12059711990921198891199103)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119910312059711990911198891199103
+ 2119890minus2119891 sin (1199093)120597119891
1205971199103
120597119871
12059711990911198891199103
minus 119890minus2119891 cos (1199093)
1205972119871
120597119910312059711990921198891199103
+ 2119890minus2119891 cos (1199093)120597119891
1205971199103
120597119871
12059711990921198891199103)+119884
3(minus
1205972119871
120597119910312059711990931198891199103
+ 2119890minus2119891120597119891
1205971199103
120597
12059711990931198891199103)minus
120597119871
12059711991031198891199103
(18)
Using (11) we get first equations as follows
1198831[minus119890
2119891 cos (1199093)1205972119871
120597119909112059711991011198891199091
minus 11989021198912
120597119891
1205971199091cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119909112059711991021198891199091
minus 11989021198912
120597119891
1205971199091sin (1199093)
120597119871
12059711991021198891199091]
+1198832[minus119890
2119891 cos (1199093)1205972119871
120597119909212059711991011198891199091
minus 11989021198912
120597119891
1205971199092cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119909212059711991021198891199091
minus 11989021198912
120597119891
1205971199092sin (1199093)
120597119871
12059711991021198891199091]
+1198833[minus119890
2119891 cos (1199093)1205972119871
120597119909312059711991011198891199091
minus 11989021198912
120597119891
1205971199093cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119909312059711991021198891199091
minus 11989021198912
120597119891
1205971199093sin (1199093)
120597119871
12059711991021198891199091]
+1198841[minus119890
2119891 cos (1199093)1205972119871
120597119910211198891199091
minus 11989021198912
120597119891
1205971199101cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119910112059711991021198891199091
minus 11989021198912
120597119891
1205971199101sin (1199093)
120597119871
12059711991021198891199091]
+1198842[minus119890
2119891 cos (1199093)1205972119871
120597119910212059711991011198891199091
minus 11989021198912
120597119891
1205971199102cos (1199093)
120597119871
12059711991011198891199091 minus 119890
2119891 sin (1199093)1205972119871
120597119910221198891199091
minus 11989021198912
120597119891
1205971199102sin (1199093)
120597119871
12059711991021198891199091]
+1198843[minus119890
2119891 cos (1199093)1205972119871
120597119910312059711991011198891199091
minus 11989021198912
120597119891
1205971199103cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119910312059711991021198891199091
minus 11989021198912
120597119891
1205971199103sin (1199093)
120597119871
12059711991021198891199091] = minus
120597119871
12059711990911198891199091
(19)
From here
minus cos (1199093) 119881(1198902119891 120597119871
1205971199101)minus sin (1199093) 119881(119890
2119891 120597119871
1205971199102)+
120597119871
1205971199091
= 0
(20)
Advances in Mathematical Physics 9
If we think of the curve 120572 for all equations as an integralcurve of 119881 that is 119881(120572) = (120597120597119905)(120572) we find the followingequations
(PDE1) minus cos (1199093)120597
120597119905(119890
2119891 120597119871
1205971199101)
minus sin (1199093)120597
120597119905(119890
2119891 120597119871
1205971199102)+
120597119871
1205971199091= 0
(PDE2) sin (1199093)120597
120597119905(119890
2119891 120597119871
1205971199101)
minus cos (1199093)120597
120597119905(119890
2119891 120597119871
1205971199102)+
120597119871
1205971199092= 0
(PDE3) minus120597
120597119905(119890
2119891 120597119871
1205971199103)+
120597119871
1205971199093= 0
(PDE4) cos (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199091)
minus sin (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199092)+
120597119871
1205971199101= 0
(PDE5) sin (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199091)
+ cos (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199092)+
120597119871
1205971199102= 0
(PDE6) 120597
120597119905(119890minus2119891 120597119871
1205971199093)+
120597119871
1205971199103= 0
(21)
such that the differential equations (21) are named conformalEuler-Lagrange equations on conformally flat manifold whichis shown in the form of (119879119872 119892 nabla 119869) Also therefore the triple(119879119872Φ
119871 119881) is called a conformal-Lagrangian mechanical
system on (119879119872 119892 nabla 119869)
7 Weyl-Euler-Lagrangian Equations forConservative Dynamical Systems
Proposition 26 We choose 119865 = i119881 119892 = Φ
119871 and 120582 = 2119891
at (11) and by considering (4) we can write Weyl-Lagrangiandynamic equation as follows
i119881(1198902119891
Φ119871) = i119881(Φ119871) minus 119889 (2119891) (22)
The second part (11) according to the law of conservation ofenergy [32] will not change for conservative dynamical systemsand i119881(Φ119871) = Φ
119871(119881)
Φ119871(119881) minus 2119889119891 = 119889119864
119871
Φ119871(119881) = 119889119864
119871+ 2119889119891 = 119889 (119864
119871+ 2119891)
(23)
From (21) above 119871 rarr 119871 + 2119891 So we can write
(PDE7) minus cos (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199101
)
minus sin (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199102
)
+120597 (119871 + 2119891)
1205971199091= 0
(PDE8) sin (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199101
)
minus cos (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199102
)
+120597 (119871 + 2119891)
1205971199092= 0
(PDE9) minus120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199103
)+120597 (119871 + 2119891)
1205971199093= 0
(PDE10) cos (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199091)
minus sin (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199092)
+120597 (119871 + 2119891)
1205971199101= 0
(PDE11) sin (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199091)
+ cos (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199092)
+120597 (119871 + 2119891)
1205971199102= 0
(PDE12) 120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199093)+
120597 (119871 + 2119891)1205971199103
= 0
(24)
and these differential equations (24) are named Weyl-Euler-Lagrange equations for conservative dynamical systems whichare constructed on conformally flat manifold (119879119872 119892 nabla 119869 119865)
and therefore the triple (119879119872Φ119871 119881) is called a Weyl-
Lagrangian mechanical system
8 Equations Solving with Computer
Theequations systems (21) and (24) have been solved by usingthe symbolic Algebra software and implicit solution is below
119871 (1199091 1199092 1199093 1199101 1199102 1199103 119905)
= exp (minus119894 lowast 119905) lowast 1198651 (1199103 minus 119894 lowast 1199093) + 1198652 (119905)
+ exp (119905 lowast 119894) lowast 1198653 (1199103 +1199093 lowast 119894) for 119891 = 0
(25)
10 Advances in Mathematical Physics
05
1
05 1 15 2minus1
minus1
minus05
minus05
(a)
05
1
05 1 15 2minus1
minus1
minus05
minus05
(b)
Figure 1
It is well known that an electromagnetic field is a physical fieldproduced by electrically charged objects The movement ofobjects in electrical magnetic and gravitational fields force isvery important For instance on a weather map the surfacewind velocity is defined by assigning a vector to each pointon a map So each vector represents the speed and directionof the movement of air at that point
The location of each object in space is represented bythree dimensions in physical space The dimensions whichare represented by higher dimensions are time positionmass and so forth The number of dimensions of (25) will bereduced to three and behind the graphics will be drawn Firstimplicit function at (25) will be selected as special After thefigure of (25) has been drawn for the route of the movementof objects in the electromagnetic field
Example 27 Consider
119871 (1199091 1199092 1199093 1199101 1199102 1199103 119905) = exp (minus119894 lowast 119905) + exp (119905 lowast 119894) lowast 119905 minus 1199052 (26)
(see Figure 1)
9 Discussion
A classical field theory explains the study of how one or morephysical fields interact with matter which is used in quantumand classical mechanics of physics branches In this study theEuler-Lagrange mechanical equations (21) and (24) derivedon a generalized on flat manifolds may be suggested to dealwith problems in electrical magnetic and gravitational fieldsforce for the path of movement (26) of defined space movingobjects [24]
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the agency BAP of PamukkaleUniversity
References
[1] W K Liu and S Jun Computational Nanomechanics of Mate-rials American Scientific Publishers Stevenson Ranch CalifUSA 2005
[2] M Tekkoyun and Y Yayli ldquoMechanical systems on generalized-quaternionic Kahler manifoldsrdquo International Journal of Geo-metric Methods in Modern Physics vol 8 no 7 pp 1419ndash14312011
[3] D B Fogel Epistemology of a theory of everything Weyl Ein-stein and the unification of physics [PhD thesis] GraduateSchool of theUniversity ofNotreDameNotreDame Ind USA2008
[4] Z Kasap and M Tekkoyun ldquoMechanical systems on almostparapseudo-KahlerndashWeyl manifoldsrdquo International Journal ofGeometric Methods in Modern Physics vol 10 no 5 Article ID1350008 8 pages 2013
[5] Z Kasap ldquoWeyl-mechanical systems on tangent manifoldsof constant 119882-sectional curvaturerdquo International Journal ofGeometric Methods in Modern Physics vol 10 no 10 Article ID1350053 pp 1ndash13 2013
[6] M Kapovich ldquoFlat conformal structures on 3-manifolds I uni-formization of closed seifert manifoldsrdquo Journal of DifferentialGeometry vol 38 no 1 pp 191ndash215 1993
[7] F Schwartz ldquoA volumetric Penrose inequality for conformallyflat manifoldsrdquo Annales Henri Poincare vol 12 no 1 pp 67ndash762011
[8] R S Kulkarni ldquoConformally flat manifoldsrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 69 pp 2675ndash2676 1972
[9] I G Dotti and R J Miatello ldquoOn the cohomology ring offlat manifolds with a special structurerdquo Revista De La Uni OnMatematica Argentina vol 46 no 2 pp 133ndash147 2005
Advances in Mathematical Physics 11
[10] A Szczepanski Kahler at Manifolds of Low Dimensions InstitutdesHautes Etudes Scientifiques Bures-sur-Yvette France 2005
[11] R Bartnik ldquoThemass of an asymptotically flat manifoldrdquo Com-munications on Pure and AppliedMathematics vol 39 no 5 pp661ndash693 1986
[12] M D M Gonzalez ldquoSingular sets of a class of locally confor-mally flat manifoldsrdquo Duke Mathematical Journal vol 129 no3 pp 551ndash572 2005
[13] S Akbulut and M Kalafat ldquoA class of locally conformally flat4-manifoldsrdquoNewYork Journal of Mathematics vol 18 pp 733ndash763 2012
[14] S-H Zhu ldquoThe classification of complete locally conformallyflat manifolds of nonnegative Ricci curvaturerdquo Pacific Journalof Mathematics vol 163 no 1 pp 189ndash199 1994
[15] H M Abood ldquoAlmost Hermitian manifold with flat Bochnertensorrdquo European Journal of Pure and Applied Mathematics vol3 no 4 pp 730ndash736 2010
[16] K Olszak and Z Olszak ldquoOn 4-dimensional conformally flatalmost 120576-Kahlerianmanifoldsrdquo Journal of Geometry and Physicsvol 62 no 5 pp 1108ndash1113 2012
[17] S Upadhyay ldquoA bounding question for almost flat manifoldsrdquoTransactions of the AmericanMathematical Society vol 353 no3 pp 963ndash972 2001
[18] 2015 httpmathworldwolframcomFlatManifoldhtml[19] 2015 httpenwikipediaorgwikiConformally flat manifold[20] D McDu and D Salamon J-Holomorphic Curves and Quantum
Cohomology AMS 1995[21] 2015 httpenwikipediaorgwikiConformal class[22] G B Folland ldquoWeyl manifoldsrdquo Journal of Differential Geome-
try vol 4 pp 145ndash153 1970[23] L Kadosh Topics in weyl geometry [PhD thesis] University of
California Berkeley Calif USA 1996[24] H Weyl Space-Time-Matter Dover Publications 1922 Trans-
lated from the 4th German edition by H Brose MethuenLondon UK Dover Publications New York NY USA 1952
[25] P Gilkey and S Nikcevic ldquoKahler and para-Kahler curvatureWeyl manifoldsrdquo httparxivorgabs10114844
[26] H Pedersen Y S Poon and A Swann ldquoThe Einstein-Weylequations in complex and quaternionic geometryrdquo DifferentialGeometry and Its Applications vol 3 no 4 pp 309ndash321 1993
[27] P Gilkey and S Nikcevic ldquoKahler-Weylmanifolds of dimension4rdquo httparxivorgabs11094532
[28] M Brozos-Vazquez P Gilkey and E Merino ldquoGeometricrealizations of Kaehler and of para-Kaehler curvature modelsrdquoInternational Journal of Geometric Methods in Modern Physicsvol 7 no 3 pp 505ndash515 2010
[29] 2015 httpenwikipediaorgwikiWeyl tensor[30] R Miron D Hrimiuc H Shimada and S V SabauThe Geom-
etry of Hamilton and Lagrange Spaces Kluwer Academic Pub-lishers 2002
[31] J Klein ldquoEspaces variationnels et mecaniquerdquo Annales delrsquoInstitut Fourier vol 12 pp 1ndash124 1962
[32] M de Leon and P R RodriguesMethods of Differential Geom-etry in Analytical Mechanics North-Holland Elsevier Amster-dam The Netherlands 1989
[33] R Abraham J E Marsden and T Ratiu Manifolds TensorAnalysis and Applications Springer New York NY USA 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Advances in Mathematical Physics
Weyl manifolds (119879119872 119892 nabla 119869) Now 119869 denotes the structure ofthe holomorphic property
1198692 120597
1205971199091= 119869 ∘ 119869
120597
1205971199091= 119890
2119891 cos (1199093) 119869120597
1205971199101+ 119890
2119891 sin (1199093) 119869
sdot120597
1205971199102= 119890
2119891 cos (1199093)
sdot [minus119890minus2119891 cos (1199093)
120597
1205971199091+ 119890minus2119891 sin (1199093)
120597
1205971199092]+ 119890
2119891
sdot sin (1199093) [minus119890minus2119891 sin (1199093)
120597
1205971199091minus 119890minus2119891 cos (1199093)
120597
1205971199092]
= minus cos2 (1199093)120597
1205971199091+ cos (1199093) sin (1199093)
120597
1205971199092
minus sin2 (1199093)120597
1205971199091minus sin (1199093) cos (1199093)
120597
1205971199092
= minus [cos2 (1199093) + sin2(1199093)]
120597
1205971199091= minus
120597
1205971199091
(9)
and in similar manner it is shown that
1198692 120597
120597119909119894
= minus120597
120597119909119894
1198692 120597
120597119910119894
= minus120597
120597119910119894
119894 = 1 2 3
(10)
As can be seen from (9) and (10) 1198692 = minus119868 are the complexstructures
5 Euler-Lagrange Dynamics Equations
Definition 24 (see [31ndash33]) Let119872 be an 119899-dimensional man-ifold and 119879119872 its tangent bundle with canonical projection120591119872 119879119872 rarr 119872 119879119872 is called the phase space of velocities of
the base manifold119872 Let 119871 119879119872 rarr R be a differentiablefunction on 119879119872 and it is called the Lagrangian function Weconsider closed 2-form on 119879119872 and Φ
119871= minus119889d
119869119871 Consider
the equation
i119881Φ119871= 119889119864119871 (11)
where the semispray 119881 is a vector field Also i is a reducingfunction and i
119881Φ119871= Φ119871(119881) We will see that for motion in
a potential 119864119871= V(119871) minus 119871 is an energy function (119871 = 119879minus119875 =
(12)119898V2 minus 119898119892ℎ kinetic-potential energies) and V = 119869119881
a Liouville vector field Here 119889119864119871denotes the differential
of 119864 We will see that (11) under a certain condition on 119881is the intrinsic expression of the Euler-Lagrange equationsof motion This equation is named Euler-Lagrange dynam-ical equation The triple (119879119872Φ
119871 119881) is known as Euler-
Lagrangian systemon the tangent bundle119879119872Theoperationsrun on (11) for any coordinate system (119902
119894
(119905) 119901119894(119905)) Infinite
dimension Lagrangianrsquos equation is obtained in the formbelow
119889
119889119905(120597119871
120597 119902119894)minus
120597119871
120597119902119894= 0
119889119902119894
119889119905= 119902119894
119894 = 1 119899
(12)
6 Conformal Weyl-Euler-LagrangianEquations
Here we using (11) obtain Weyl-Euler-Lagrange equationsfor classical and quantum mechanics on conformally flatmanifold and it is shown by (119879119872 119892 nabla 119869)
Proposition 25 Let (119909119894 119910119894) be coordinate functions Also on
(119879119872 119892 nabla 119869) let 119881 be the vector field determined by 119881 =
sum3119894=1(119883119894
(120597120597119909119894) + 119884119894
(120597120597119910119894)) Then the vector field defined by
V = 119869119881
= 1198831(119890
2119891 cos (1199093)120597
1205971199101+ 119890
2119891 sin (1199093)120597
1205971199102)
+1198832(minus119890
2119891 sin (1199093)120597
1205971199101+ 119890
2119891 cos (1199093)120597
1205971199102)
+11988331198902119891 120597
1205971199103
+1198841(minus119890minus2119891 cos (1199093)
120597
1205971199091+ 119890minus2119891 sin (1199093)
120597
1205971199092)
+1198842(minus119890minus2119891 sin (1199093)
120597
1205971199091minus 119890minus2119891 cos (1199093)
120597
1205971199092)
minus1198843119890minus2119891 120597
1205971199093
(13)
is thought to be Weyl-Liouville vector field on conformally flatmanifold (119879119872 119892 nabla 119869) Φ
119871= minus119889d
119869119871 is the closed 2-form
given by (11) such that d = sum3119894=1((120597120597119909119894)119889119909119894 + (120597120597119910119894)119889119910119894)
d119869 119865(119872) rarr and
1119872 d119869= 119894119869d minus d119894
119869 and d
119869= 119869(d) =
sum3119894=1(119883119894
119869(120597120597119909119894)+119884119894
119869(120597120597119910119894)) Also the vertical differentiation
d119869is given where 119889 is the usual exterior derivationThen there
is the following result We can obtain Weyl-Euler-Lagrangeequations for classical and quantummechanics on conformallyflat manifold (119879119872 119892 nabla 119869) We get the equations given by
d119869= [119890
2119891 cos (1199093)120597
1205971199101+ 119890
2119891 sin (1199093)120597
1205971199102] 1198891199091
+[minus1198902119891 sin (1199093)
120597
1205971199101+ 119890
2119891 cos (1199093)120597
1205971199102] 1198891199092
+ 1198902119891 120597
12059711991031198891199093
Advances in Mathematical Physics 5
+[minus119890minus2119891 cos (1199093)
120597
1205971199091+ 119890minus2119891 sin (1199093)
120597
1205971199092] 1198891199101
+[minus119890minus2119891 sin (1199093)
120597
1205971199091minus 119890minus2119891 cos (1199093)
120597
1205971199092] 1198891199102
minus 119890minus2119891 120597
12059711990931198891199103
(14)
Also
Φ119871= minus119889d
119869119871
= minus119889([1198902119891 cos (1199093)
120597
1205971199101+ 119890
2119891 sin (1199093)120597
1205971199102] 1198891199091
+[minus1198902119891 sin (1199093)
120597
1205971199101+ 119890
2119891 cos (1199093)120597
1205971199102] 1198891199092
+ 1198902119891 120597119871
12059711991031198891199093
+[minus119890minus2119891 cos (1199093)
120597
1205971199091+ 119890minus2119891 sin (1199093)
120597
1205971199092] 1198891199101
+[minus119890minus2119891 sin (1199093)
120597
1205971199091minus 119890minus2119891 cos (1199093)
120597
1205971199092] 1198891199102
minus 119890minus2119891 120597119871
12059711990931198891199103)
(15)
and then we find
i119881Φ119871= Φ119871(119881) = Φ
119871(
3sum
119894=1(119883119894120597
120597119909119894
+119884119894120597
120597119910119894
)) (16)
Moreover the energy function of system is
119864119871= 119883
1[119890
2119891 cos (1199093)120597119871
1205971199101+ 119890
2119891 sin (1199093)120597119871
1205971199102]
+1198832[minus119890
2119891 sin (1199093)120597119871
1205971199101+ 119890
2119891 cos (1199093)120597119871
1205971199102]
+11988331198902119891 120597119871
1205971199103
+1198841[minus119890minus2119891 cos (1199093)
120597119871
1205971199091+ 119890minus2119891 sin (1199093)
120597119871
1205971199092]
+1198842[minus119890minus2119891 sin (1199093)
120597119871
1205971199091minus 119890minus2119891 cos (1199093)
120597119871
1205971199092]
minus1198843119890minus2119891 120597119871
1205971199093minus119871
(17)
and the differential of 119864119871is
119889119864119871= 119883
1(119890
2119891 cos (1199093)1205972119871
120597119909112059711991011198891199091
+ 21198902119891 cos (1199093)120597119891
1205971199091
120597119871
12059711991011198891199091
+ 1198902119891 sin (1199093)
1205972119871
120597119909112059711991021198891199091
+ 21198902119891 sin (1199093)120597119891
1205971199091
120597119871
12059711991021198891199091)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119909112059711991011198891199091
minus 21198902119891 sin (1199093)120597119891
1205971199091
120597119871
12059711991011198891199091
+ 1198902119891 cos (1199093)
1205972119871
120597119909112059711991021198891199091
+ 21198902119891 cos (1199093)120597119891
1205971199091
120597119871
12059711991021198891199091)
+1198833(119890
2119891 1205972119871
120597119909112059711991031198891199091 + 2119890
2119891 120597119891
1205971199091
120597119871
12059711991031198891199091)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119909211198891199091
+ 2119890minus2119891 cos (1199093)120597119891
1205971199091
120597119871
12059711990911198891199091
+ 119890minus2119891 sin (1199093)
1205972119871
120597119909112059711990921198891199091
minus 2119890minus2119891 sin (1199093)120597119891
1205971199091
120597119871
12059711990921198891199091)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119909211198891199091
+ 2119890minus2119891 sin (1199093)120597119891
1205971199091
120597119871
12059711990911198891199091
minus 119890minus2119891 cos (1199093)
1205972119871
120597119909112059711990921198891199091
+ 2119890minus2119891 cos (1199093)120597119891
1205971199091
120597119871
12059711990921198891199091)
+1198843(minus
1205972119871
120597119909112059711990931198891199091 + 2119890
minus2119891 120597119891
1205971199091
120597
12059711990931198891199091)
minus120597119871
12059711990911198891199091 +119883
1(119890
2119891 cos (1199093)1205972119871
120597119909212059711991011198891199092
6 Advances in Mathematical Physics
+ 21198902119891 cos (1199093)120597119891
1205971199092
120597119871
12059711991011198891199092
+ 1198902119891 sin (1199093)
1205972119871
120597119909212059711991021198891199092
+ 21198902119891 sin (1199093)120597119891
1205971199092
120597119871
12059711991021198891199092)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119909212059711991011198891199092
minus 21198902119891 sin (1199093)120597119891
1205971199092
120597119871
12059711991011198891199092
+ 1198902119891 cos (1199093)
1205972119871
120597119909212059711991021198891199092
+ 21198902119891 cos (1199093)120597119891
1205971199092
120597119871
12059711991021198891199092)
+1198833(119890
2119891 1205972119871
120597119909212059711991031198891199092 + 2119890
2119891 120597119891
1205971199092
120597119871
12059711991031198891199092)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119909212059711990911198891199092
+ 2119890minus2119891 cos (1199093)120597119891
1205971199092
120597119871
12059711990911198891199092
+ 119890minus2119891 sin (1199093)
1205972119871
120597119909221198891199092
minus 2119890minus2119891 sin (1199093)120597119891
1205971199092
120597119871
12059711990921198891199092)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119909212059711990911198891199092
+ 2119890minus2119891 sin (1199093)120597119891
1205971199092
120597119871
12059711990911198891199092
minus 119890minus2119891 cos (1199093)
1205972119871
120597119909221198891199092
+ 2119890minus2119891 cos (1199093)120597119891
1205971199092
120597119871
12059711990921198891199092)
+1198843(minus
1205972119871
120597119909212059711990931198891199092 + 2119890
minus2119891 120597119891
1205971199092
120597
12059711990931198891199092)
minus120597119871
12059711990921198891199092 +119883
1(119890
2119891 cos (1199093)1205972119871
120597119909312059711991011198891199093
+ 21198902119891 cos (1199093)120597119891
1205971199093
120597119871
12059711991011198891199093 minus 119890
2119891 sin (1199093)120597119871
12059711991011198891199093
+ 1198902119891 sin (1199093)
1205972119871
120597119909312059711991021198891199093
+ 21198902119891 sin (1199093)120597119891
1205971199093
120597119871
12059711991021198891199093
+ 1198902119891 cos (1199093)
120597119871
12059711991021198891199093)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119909312059711991011198891199093
minus 21198902119891 sin (1199093)120597119891
1205971199093
120597119871
12059711991011198891199093 minus 119890
2119891 cos (1199093)120597119871
12059711991011198891199093
+ 1198902119891 cos (1199093)
1205972119871
120597119909312059711991021198891199093
+ 21198902119891 cos (1199093)120597119891
1205971199093
120597119871
12059711991021198891199093
minus 1198902119891 sin (1199093)
120597119871
12059711991021198891199093)+119883
3(119890
2119891 1205972119871
120597119909312059711991031198891199093
+ 21198902119891120597119891
1205971199093
120597119871
12059711991031198891199093)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119909312059711990911198891199093
+ 2119890minus2119891 cos (1199093)120597119891
1205971199093
120597119871
12059711990911198891199093
minus 119890minus2119891 sin (1199093)
120597119871
12059711990911198891199093 + 119890
minus2119891 sin (1199093)1205972119871
120597119909312059711990921198891199093
minus 2119890minus2119891 sin (1199093)120597119891
1205971199093
120597119871
12059711990921198891199093
+ 119890minus2119891 cos (1199093)
120597119871
12059711990921198891199093)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119909312059711990911198891199093
+ 2119890minus2119891 sin (1199093)120597119891
1205971199093
120597119871
12059711990911198891199093
minus 119890minus2119891 cos (1199093)
120597119871
12059711990911198891199093
minus 119890minus2119891 cos (1199093)
1205972119871
120597119909312059711990921198891199093
+ 2119890minus2119891 cos (1199093)120597119891
1205971199093
120597119871
12059711990921198891199093
+ 119890minus2119891 sin (1199093)
120597119871
12059711990921198891199093)+119884
3(minus
1205972119871
120597119909231198891199093
+ 2119890minus2119891120597119891
1205971199093
120597
12059711990931198891199093)minus
120597119871
12059711990931198891199093
Advances in Mathematical Physics 7
+1198831(119890
2119891 cos (1199093)1205972119871
120597119910211198891199101
+ 21198902119891 cos (1199093)120597119891
1205971199101
120597119871
12059711991011198891199101
+ 1198902119891 sin (1199093)
1205972119871
120597119910112059711991021198891199101
+ 21198902119891 sin (1199093)120597119891
1205971199101
120597119871
12059711991021198891199101)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119910211198891199101
minus 21198902119891 sin (1199093)120597119891
1205971199101
120597119871
12059711991011198891199101
+ 1198902119891 cos (1199093)
1205972119871
120597119910112059711991021198891199101
+ 21198902119891 cos (1199093)120597119891
1199101
120597119871
12059711991021198891199101)+119883
3(119890
2119891 1205972119871
120597119910112059711991031198891199101
+ 21198902119891120597119891
1205971199101
120597119871
12059711991031198891199101)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119910112059711990911198891199101
+ 2119890minus2119891 cos (1199093)120597119891
1205971199101
120597119871
12059711990911198891199101
+ 119890minus2119891 sin (1199093)
1205972119871
120597119910112059711990921198891199101
minus 2119890minus2119891 sin (1199093)120597119891
1205971199101
120597119871
12059711990921198891199101)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119910112059711990911198891199101
+ 2119890minus2119891 sin (1199093)120597119891
1205971199101
120597119871
12059711990911198891199101
minus 119890minus2119891 cos (1199093)
1205972119871
120597119910112059711990921198891199101
+ 2119890minus2119891 cos (1199093)120597119891
1205971199101
120597119871
12059711990921198891199101)+119884
3(minus
1205972119871
120597119910112059711990931198891199101
+ 2119890minus2119891120597119891
1205971199101
120597
12059711990931198891199101)minus
120597119871
12059711991011198891199101
+1198831(119890
2119891 cos (1199093)1205972119871
120597119910212059711991011198891199102
+ 21198902119891 cos (1199093)120597119891
1205971199102
120597119871
12059711991011198891199102 + 119890
2119891 sin (1199093)1205972119871
120597119910221198891199102
+ 21198902119891 sin (1199093)120597119891
1205971199102
120597119871
12059711991021198891199102)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119910212059711991011198891199102
minus 21198902119891 sin (1199093)120597119891
1205971199102
120597119871
12059711991011198891199102 + 119890
2119891 cos (1199093)1205972119871
120597119910221198891199102
+ 21198902119891 cos (1199093)120597119891
1205971199102
120597119871
12059711991021198891199102)
+1198833(119890
2119891 1205972119871
120597119910212059711991031198891199102 + 2119890
2119891 120597119891
1205971199102
120597119871
12059711991031198891199102)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119910212059711990911198891199102
+ 2119890minus2119891 cos (1199093)120597119891
1205971199102
120597119871
12059711990911198891199102
+ 119890minus2119891 sin (1199093)
1205972119871
120597119910212059711990921198891199102
minus 2119890minus2119891 sin (1199093)120597119891
1205971199102
120597119871
12059711990921198891199102)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119910212059711990911198891199102
+ 2119890minus2119891 sin (1199093)120597119891
1205971199102
120597119871
12059711990911198891199102
minus 119890minus2119891 cos (1199093)
1205972119871
120597119910212059711990921198891199102
+ 2119890minus2119891 cos (1199093)120597119891
1205971199102
120597119871
12059711990921198891199102)+119884
3(minus
1205972119871
120597119910212059711990931198891199102
+ 2119890minus2119891120597119891
1205971199102
120597
12059711990931198891199102)minus
120597119871
12059711991021198891199102
+1198831(119890
2119891 cos (1199093)1205972119871
120597119910312059711991011198891199103
+ 21198902119891 cos (1199093)120597119891
1205971199103
120597119871
12059711991011198891199103
+ 1198902119891 sin (1199093)
1205972119871
120597119910312059711991021198891199103
+ 21198902119891 sin (1199093)120597119891
1205971199103
120597119871
12059711991021198891199103)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119910312059711991011198891199103
minus 21198902119891 sin (1199093)120597119891
1205971199103
120597119871
12059711991011198891199103
8 Advances in Mathematical Physics
+ 1198902119891 cos (1199093)
1205972119871
120597119910312059711991021198891199103
+ 21198902119891 cos (1199093)120597119891
1205971199103
120597119871
12059711991021198891199103)+119883
3(119890
2119891 1205972119871
120597119910231198891199103
+ 21198902119891120597119891
1205971199103
120597119871
12059711991031198891199103)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119910312059711990911198891199103
+ 2119890minus2119891 cos (1199093)120597119891
1205971199103
120597119871
12059711990911198891199103
+ 119890minus2119891 sin (1199093)
1205972119871
120597119910312059711990921198891199103
minus 2119890minus2119891 sin (1199093)120597119891
1205971199103
120597119871
12059711990921198891199103)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119910312059711990911198891199103
+ 2119890minus2119891 sin (1199093)120597119891
1205971199103
120597119871
12059711990911198891199103
minus 119890minus2119891 cos (1199093)
1205972119871
120597119910312059711990921198891199103
+ 2119890minus2119891 cos (1199093)120597119891
1205971199103
120597119871
12059711990921198891199103)+119884
3(minus
1205972119871
120597119910312059711990931198891199103
+ 2119890minus2119891120597119891
1205971199103
120597
12059711990931198891199103)minus
120597119871
12059711991031198891199103
(18)
Using (11) we get first equations as follows
1198831[minus119890
2119891 cos (1199093)1205972119871
120597119909112059711991011198891199091
minus 11989021198912
120597119891
1205971199091cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119909112059711991021198891199091
minus 11989021198912
120597119891
1205971199091sin (1199093)
120597119871
12059711991021198891199091]
+1198832[minus119890
2119891 cos (1199093)1205972119871
120597119909212059711991011198891199091
minus 11989021198912
120597119891
1205971199092cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119909212059711991021198891199091
minus 11989021198912
120597119891
1205971199092sin (1199093)
120597119871
12059711991021198891199091]
+1198833[minus119890
2119891 cos (1199093)1205972119871
120597119909312059711991011198891199091
minus 11989021198912
120597119891
1205971199093cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119909312059711991021198891199091
minus 11989021198912
120597119891
1205971199093sin (1199093)
120597119871
12059711991021198891199091]
+1198841[minus119890
2119891 cos (1199093)1205972119871
120597119910211198891199091
minus 11989021198912
120597119891
1205971199101cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119910112059711991021198891199091
minus 11989021198912
120597119891
1205971199101sin (1199093)
120597119871
12059711991021198891199091]
+1198842[minus119890
2119891 cos (1199093)1205972119871
120597119910212059711991011198891199091
minus 11989021198912
120597119891
1205971199102cos (1199093)
120597119871
12059711991011198891199091 minus 119890
2119891 sin (1199093)1205972119871
120597119910221198891199091
minus 11989021198912
120597119891
1205971199102sin (1199093)
120597119871
12059711991021198891199091]
+1198843[minus119890
2119891 cos (1199093)1205972119871
120597119910312059711991011198891199091
minus 11989021198912
120597119891
1205971199103cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119910312059711991021198891199091
minus 11989021198912
120597119891
1205971199103sin (1199093)
120597119871
12059711991021198891199091] = minus
120597119871
12059711990911198891199091
(19)
From here
minus cos (1199093) 119881(1198902119891 120597119871
1205971199101)minus sin (1199093) 119881(119890
2119891 120597119871
1205971199102)+
120597119871
1205971199091
= 0
(20)
Advances in Mathematical Physics 9
If we think of the curve 120572 for all equations as an integralcurve of 119881 that is 119881(120572) = (120597120597119905)(120572) we find the followingequations
(PDE1) minus cos (1199093)120597
120597119905(119890
2119891 120597119871
1205971199101)
minus sin (1199093)120597
120597119905(119890
2119891 120597119871
1205971199102)+
120597119871
1205971199091= 0
(PDE2) sin (1199093)120597
120597119905(119890
2119891 120597119871
1205971199101)
minus cos (1199093)120597
120597119905(119890
2119891 120597119871
1205971199102)+
120597119871
1205971199092= 0
(PDE3) minus120597
120597119905(119890
2119891 120597119871
1205971199103)+
120597119871
1205971199093= 0
(PDE4) cos (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199091)
minus sin (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199092)+
120597119871
1205971199101= 0
(PDE5) sin (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199091)
+ cos (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199092)+
120597119871
1205971199102= 0
(PDE6) 120597
120597119905(119890minus2119891 120597119871
1205971199093)+
120597119871
1205971199103= 0
(21)
such that the differential equations (21) are named conformalEuler-Lagrange equations on conformally flat manifold whichis shown in the form of (119879119872 119892 nabla 119869) Also therefore the triple(119879119872Φ
119871 119881) is called a conformal-Lagrangian mechanical
system on (119879119872 119892 nabla 119869)
7 Weyl-Euler-Lagrangian Equations forConservative Dynamical Systems
Proposition 26 We choose 119865 = i119881 119892 = Φ
119871 and 120582 = 2119891
at (11) and by considering (4) we can write Weyl-Lagrangiandynamic equation as follows
i119881(1198902119891
Φ119871) = i119881(Φ119871) minus 119889 (2119891) (22)
The second part (11) according to the law of conservation ofenergy [32] will not change for conservative dynamical systemsand i119881(Φ119871) = Φ
119871(119881)
Φ119871(119881) minus 2119889119891 = 119889119864
119871
Φ119871(119881) = 119889119864
119871+ 2119889119891 = 119889 (119864
119871+ 2119891)
(23)
From (21) above 119871 rarr 119871 + 2119891 So we can write
(PDE7) minus cos (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199101
)
minus sin (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199102
)
+120597 (119871 + 2119891)
1205971199091= 0
(PDE8) sin (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199101
)
minus cos (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199102
)
+120597 (119871 + 2119891)
1205971199092= 0
(PDE9) minus120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199103
)+120597 (119871 + 2119891)
1205971199093= 0
(PDE10) cos (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199091)
minus sin (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199092)
+120597 (119871 + 2119891)
1205971199101= 0
(PDE11) sin (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199091)
+ cos (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199092)
+120597 (119871 + 2119891)
1205971199102= 0
(PDE12) 120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199093)+
120597 (119871 + 2119891)1205971199103
= 0
(24)
and these differential equations (24) are named Weyl-Euler-Lagrange equations for conservative dynamical systems whichare constructed on conformally flat manifold (119879119872 119892 nabla 119869 119865)
and therefore the triple (119879119872Φ119871 119881) is called a Weyl-
Lagrangian mechanical system
8 Equations Solving with Computer
Theequations systems (21) and (24) have been solved by usingthe symbolic Algebra software and implicit solution is below
119871 (1199091 1199092 1199093 1199101 1199102 1199103 119905)
= exp (minus119894 lowast 119905) lowast 1198651 (1199103 minus 119894 lowast 1199093) + 1198652 (119905)
+ exp (119905 lowast 119894) lowast 1198653 (1199103 +1199093 lowast 119894) for 119891 = 0
(25)
10 Advances in Mathematical Physics
05
1
05 1 15 2minus1
minus1
minus05
minus05
(a)
05
1
05 1 15 2minus1
minus1
minus05
minus05
(b)
Figure 1
It is well known that an electromagnetic field is a physical fieldproduced by electrically charged objects The movement ofobjects in electrical magnetic and gravitational fields force isvery important For instance on a weather map the surfacewind velocity is defined by assigning a vector to each pointon a map So each vector represents the speed and directionof the movement of air at that point
The location of each object in space is represented bythree dimensions in physical space The dimensions whichare represented by higher dimensions are time positionmass and so forth The number of dimensions of (25) will bereduced to three and behind the graphics will be drawn Firstimplicit function at (25) will be selected as special After thefigure of (25) has been drawn for the route of the movementof objects in the electromagnetic field
Example 27 Consider
119871 (1199091 1199092 1199093 1199101 1199102 1199103 119905) = exp (minus119894 lowast 119905) + exp (119905 lowast 119894) lowast 119905 minus 1199052 (26)
(see Figure 1)
9 Discussion
A classical field theory explains the study of how one or morephysical fields interact with matter which is used in quantumand classical mechanics of physics branches In this study theEuler-Lagrange mechanical equations (21) and (24) derivedon a generalized on flat manifolds may be suggested to dealwith problems in electrical magnetic and gravitational fieldsforce for the path of movement (26) of defined space movingobjects [24]
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the agency BAP of PamukkaleUniversity
References
[1] W K Liu and S Jun Computational Nanomechanics of Mate-rials American Scientific Publishers Stevenson Ranch CalifUSA 2005
[2] M Tekkoyun and Y Yayli ldquoMechanical systems on generalized-quaternionic Kahler manifoldsrdquo International Journal of Geo-metric Methods in Modern Physics vol 8 no 7 pp 1419ndash14312011
[3] D B Fogel Epistemology of a theory of everything Weyl Ein-stein and the unification of physics [PhD thesis] GraduateSchool of theUniversity ofNotreDameNotreDame Ind USA2008
[4] Z Kasap and M Tekkoyun ldquoMechanical systems on almostparapseudo-KahlerndashWeyl manifoldsrdquo International Journal ofGeometric Methods in Modern Physics vol 10 no 5 Article ID1350008 8 pages 2013
[5] Z Kasap ldquoWeyl-mechanical systems on tangent manifoldsof constant 119882-sectional curvaturerdquo International Journal ofGeometric Methods in Modern Physics vol 10 no 10 Article ID1350053 pp 1ndash13 2013
[6] M Kapovich ldquoFlat conformal structures on 3-manifolds I uni-formization of closed seifert manifoldsrdquo Journal of DifferentialGeometry vol 38 no 1 pp 191ndash215 1993
[7] F Schwartz ldquoA volumetric Penrose inequality for conformallyflat manifoldsrdquo Annales Henri Poincare vol 12 no 1 pp 67ndash762011
[8] R S Kulkarni ldquoConformally flat manifoldsrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 69 pp 2675ndash2676 1972
[9] I G Dotti and R J Miatello ldquoOn the cohomology ring offlat manifolds with a special structurerdquo Revista De La Uni OnMatematica Argentina vol 46 no 2 pp 133ndash147 2005
Advances in Mathematical Physics 11
[10] A Szczepanski Kahler at Manifolds of Low Dimensions InstitutdesHautes Etudes Scientifiques Bures-sur-Yvette France 2005
[11] R Bartnik ldquoThemass of an asymptotically flat manifoldrdquo Com-munications on Pure and AppliedMathematics vol 39 no 5 pp661ndash693 1986
[12] M D M Gonzalez ldquoSingular sets of a class of locally confor-mally flat manifoldsrdquo Duke Mathematical Journal vol 129 no3 pp 551ndash572 2005
[13] S Akbulut and M Kalafat ldquoA class of locally conformally flat4-manifoldsrdquoNewYork Journal of Mathematics vol 18 pp 733ndash763 2012
[14] S-H Zhu ldquoThe classification of complete locally conformallyflat manifolds of nonnegative Ricci curvaturerdquo Pacific Journalof Mathematics vol 163 no 1 pp 189ndash199 1994
[15] H M Abood ldquoAlmost Hermitian manifold with flat Bochnertensorrdquo European Journal of Pure and Applied Mathematics vol3 no 4 pp 730ndash736 2010
[16] K Olszak and Z Olszak ldquoOn 4-dimensional conformally flatalmost 120576-Kahlerianmanifoldsrdquo Journal of Geometry and Physicsvol 62 no 5 pp 1108ndash1113 2012
[17] S Upadhyay ldquoA bounding question for almost flat manifoldsrdquoTransactions of the AmericanMathematical Society vol 353 no3 pp 963ndash972 2001
[18] 2015 httpmathworldwolframcomFlatManifoldhtml[19] 2015 httpenwikipediaorgwikiConformally flat manifold[20] D McDu and D Salamon J-Holomorphic Curves and Quantum
Cohomology AMS 1995[21] 2015 httpenwikipediaorgwikiConformal class[22] G B Folland ldquoWeyl manifoldsrdquo Journal of Differential Geome-
try vol 4 pp 145ndash153 1970[23] L Kadosh Topics in weyl geometry [PhD thesis] University of
California Berkeley Calif USA 1996[24] H Weyl Space-Time-Matter Dover Publications 1922 Trans-
lated from the 4th German edition by H Brose MethuenLondon UK Dover Publications New York NY USA 1952
[25] P Gilkey and S Nikcevic ldquoKahler and para-Kahler curvatureWeyl manifoldsrdquo httparxivorgabs10114844
[26] H Pedersen Y S Poon and A Swann ldquoThe Einstein-Weylequations in complex and quaternionic geometryrdquo DifferentialGeometry and Its Applications vol 3 no 4 pp 309ndash321 1993
[27] P Gilkey and S Nikcevic ldquoKahler-Weylmanifolds of dimension4rdquo httparxivorgabs11094532
[28] M Brozos-Vazquez P Gilkey and E Merino ldquoGeometricrealizations of Kaehler and of para-Kaehler curvature modelsrdquoInternational Journal of Geometric Methods in Modern Physicsvol 7 no 3 pp 505ndash515 2010
[29] 2015 httpenwikipediaorgwikiWeyl tensor[30] R Miron D Hrimiuc H Shimada and S V SabauThe Geom-
etry of Hamilton and Lagrange Spaces Kluwer Academic Pub-lishers 2002
[31] J Klein ldquoEspaces variationnels et mecaniquerdquo Annales delrsquoInstitut Fourier vol 12 pp 1ndash124 1962
[32] M de Leon and P R RodriguesMethods of Differential Geom-etry in Analytical Mechanics North-Holland Elsevier Amster-dam The Netherlands 1989
[33] R Abraham J E Marsden and T Ratiu Manifolds TensorAnalysis and Applications Springer New York NY USA 2001
Submit your manuscripts athttpwwwhindawicom
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 5
+[minus119890minus2119891 cos (1199093)
120597
1205971199091+ 119890minus2119891 sin (1199093)
120597
1205971199092] 1198891199101
+[minus119890minus2119891 sin (1199093)
120597
1205971199091minus 119890minus2119891 cos (1199093)
120597
1205971199092] 1198891199102
minus 119890minus2119891 120597
12059711990931198891199103
(14)
Also
Φ119871= minus119889d
119869119871
= minus119889([1198902119891 cos (1199093)
120597
1205971199101+ 119890
2119891 sin (1199093)120597
1205971199102] 1198891199091
+[minus1198902119891 sin (1199093)
120597
1205971199101+ 119890
2119891 cos (1199093)120597
1205971199102] 1198891199092
+ 1198902119891 120597119871
12059711991031198891199093
+[minus119890minus2119891 cos (1199093)
120597
1205971199091+ 119890minus2119891 sin (1199093)
120597
1205971199092] 1198891199101
+[minus119890minus2119891 sin (1199093)
120597
1205971199091minus 119890minus2119891 cos (1199093)
120597
1205971199092] 1198891199102
minus 119890minus2119891 120597119871
12059711990931198891199103)
(15)
and then we find
i119881Φ119871= Φ119871(119881) = Φ
119871(
3sum
119894=1(119883119894120597
120597119909119894
+119884119894120597
120597119910119894
)) (16)
Moreover the energy function of system is
119864119871= 119883
1[119890
2119891 cos (1199093)120597119871
1205971199101+ 119890
2119891 sin (1199093)120597119871
1205971199102]
+1198832[minus119890
2119891 sin (1199093)120597119871
1205971199101+ 119890
2119891 cos (1199093)120597119871
1205971199102]
+11988331198902119891 120597119871
1205971199103
+1198841[minus119890minus2119891 cos (1199093)
120597119871
1205971199091+ 119890minus2119891 sin (1199093)
120597119871
1205971199092]
+1198842[minus119890minus2119891 sin (1199093)
120597119871
1205971199091minus 119890minus2119891 cos (1199093)
120597119871
1205971199092]
minus1198843119890minus2119891 120597119871
1205971199093minus119871
(17)
and the differential of 119864119871is
119889119864119871= 119883
1(119890
2119891 cos (1199093)1205972119871
120597119909112059711991011198891199091
+ 21198902119891 cos (1199093)120597119891
1205971199091
120597119871
12059711991011198891199091
+ 1198902119891 sin (1199093)
1205972119871
120597119909112059711991021198891199091
+ 21198902119891 sin (1199093)120597119891
1205971199091
120597119871
12059711991021198891199091)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119909112059711991011198891199091
minus 21198902119891 sin (1199093)120597119891
1205971199091
120597119871
12059711991011198891199091
+ 1198902119891 cos (1199093)
1205972119871
120597119909112059711991021198891199091
+ 21198902119891 cos (1199093)120597119891
1205971199091
120597119871
12059711991021198891199091)
+1198833(119890
2119891 1205972119871
120597119909112059711991031198891199091 + 2119890
2119891 120597119891
1205971199091
120597119871
12059711991031198891199091)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119909211198891199091
+ 2119890minus2119891 cos (1199093)120597119891
1205971199091
120597119871
12059711990911198891199091
+ 119890minus2119891 sin (1199093)
1205972119871
120597119909112059711990921198891199091
minus 2119890minus2119891 sin (1199093)120597119891
1205971199091
120597119871
12059711990921198891199091)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119909211198891199091
+ 2119890minus2119891 sin (1199093)120597119891
1205971199091
120597119871
12059711990911198891199091
minus 119890minus2119891 cos (1199093)
1205972119871
120597119909112059711990921198891199091
+ 2119890minus2119891 cos (1199093)120597119891
1205971199091
120597119871
12059711990921198891199091)
+1198843(minus
1205972119871
120597119909112059711990931198891199091 + 2119890
minus2119891 120597119891
1205971199091
120597
12059711990931198891199091)
minus120597119871
12059711990911198891199091 +119883
1(119890
2119891 cos (1199093)1205972119871
120597119909212059711991011198891199092
6 Advances in Mathematical Physics
+ 21198902119891 cos (1199093)120597119891
1205971199092
120597119871
12059711991011198891199092
+ 1198902119891 sin (1199093)
1205972119871
120597119909212059711991021198891199092
+ 21198902119891 sin (1199093)120597119891
1205971199092
120597119871
12059711991021198891199092)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119909212059711991011198891199092
minus 21198902119891 sin (1199093)120597119891
1205971199092
120597119871
12059711991011198891199092
+ 1198902119891 cos (1199093)
1205972119871
120597119909212059711991021198891199092
+ 21198902119891 cos (1199093)120597119891
1205971199092
120597119871
12059711991021198891199092)
+1198833(119890
2119891 1205972119871
120597119909212059711991031198891199092 + 2119890
2119891 120597119891
1205971199092
120597119871
12059711991031198891199092)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119909212059711990911198891199092
+ 2119890minus2119891 cos (1199093)120597119891
1205971199092
120597119871
12059711990911198891199092
+ 119890minus2119891 sin (1199093)
1205972119871
120597119909221198891199092
minus 2119890minus2119891 sin (1199093)120597119891
1205971199092
120597119871
12059711990921198891199092)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119909212059711990911198891199092
+ 2119890minus2119891 sin (1199093)120597119891
1205971199092
120597119871
12059711990911198891199092
minus 119890minus2119891 cos (1199093)
1205972119871
120597119909221198891199092
+ 2119890minus2119891 cos (1199093)120597119891
1205971199092
120597119871
12059711990921198891199092)
+1198843(minus
1205972119871
120597119909212059711990931198891199092 + 2119890
minus2119891 120597119891
1205971199092
120597
12059711990931198891199092)
minus120597119871
12059711990921198891199092 +119883
1(119890
2119891 cos (1199093)1205972119871
120597119909312059711991011198891199093
+ 21198902119891 cos (1199093)120597119891
1205971199093
120597119871
12059711991011198891199093 minus 119890
2119891 sin (1199093)120597119871
12059711991011198891199093
+ 1198902119891 sin (1199093)
1205972119871
120597119909312059711991021198891199093
+ 21198902119891 sin (1199093)120597119891
1205971199093
120597119871
12059711991021198891199093
+ 1198902119891 cos (1199093)
120597119871
12059711991021198891199093)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119909312059711991011198891199093
minus 21198902119891 sin (1199093)120597119891
1205971199093
120597119871
12059711991011198891199093 minus 119890
2119891 cos (1199093)120597119871
12059711991011198891199093
+ 1198902119891 cos (1199093)
1205972119871
120597119909312059711991021198891199093
+ 21198902119891 cos (1199093)120597119891
1205971199093
120597119871
12059711991021198891199093
minus 1198902119891 sin (1199093)
120597119871
12059711991021198891199093)+119883
3(119890
2119891 1205972119871
120597119909312059711991031198891199093
+ 21198902119891120597119891
1205971199093
120597119871
12059711991031198891199093)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119909312059711990911198891199093
+ 2119890minus2119891 cos (1199093)120597119891
1205971199093
120597119871
12059711990911198891199093
minus 119890minus2119891 sin (1199093)
120597119871
12059711990911198891199093 + 119890
minus2119891 sin (1199093)1205972119871
120597119909312059711990921198891199093
minus 2119890minus2119891 sin (1199093)120597119891
1205971199093
120597119871
12059711990921198891199093
+ 119890minus2119891 cos (1199093)
120597119871
12059711990921198891199093)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119909312059711990911198891199093
+ 2119890minus2119891 sin (1199093)120597119891
1205971199093
120597119871
12059711990911198891199093
minus 119890minus2119891 cos (1199093)
120597119871
12059711990911198891199093
minus 119890minus2119891 cos (1199093)
1205972119871
120597119909312059711990921198891199093
+ 2119890minus2119891 cos (1199093)120597119891
1205971199093
120597119871
12059711990921198891199093
+ 119890minus2119891 sin (1199093)
120597119871
12059711990921198891199093)+119884
3(minus
1205972119871
120597119909231198891199093
+ 2119890minus2119891120597119891
1205971199093
120597
12059711990931198891199093)minus
120597119871
12059711990931198891199093
Advances in Mathematical Physics 7
+1198831(119890
2119891 cos (1199093)1205972119871
120597119910211198891199101
+ 21198902119891 cos (1199093)120597119891
1205971199101
120597119871
12059711991011198891199101
+ 1198902119891 sin (1199093)
1205972119871
120597119910112059711991021198891199101
+ 21198902119891 sin (1199093)120597119891
1205971199101
120597119871
12059711991021198891199101)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119910211198891199101
minus 21198902119891 sin (1199093)120597119891
1205971199101
120597119871
12059711991011198891199101
+ 1198902119891 cos (1199093)
1205972119871
120597119910112059711991021198891199101
+ 21198902119891 cos (1199093)120597119891
1199101
120597119871
12059711991021198891199101)+119883
3(119890
2119891 1205972119871
120597119910112059711991031198891199101
+ 21198902119891120597119891
1205971199101
120597119871
12059711991031198891199101)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119910112059711990911198891199101
+ 2119890minus2119891 cos (1199093)120597119891
1205971199101
120597119871
12059711990911198891199101
+ 119890minus2119891 sin (1199093)
1205972119871
120597119910112059711990921198891199101
minus 2119890minus2119891 sin (1199093)120597119891
1205971199101
120597119871
12059711990921198891199101)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119910112059711990911198891199101
+ 2119890minus2119891 sin (1199093)120597119891
1205971199101
120597119871
12059711990911198891199101
minus 119890minus2119891 cos (1199093)
1205972119871
120597119910112059711990921198891199101
+ 2119890minus2119891 cos (1199093)120597119891
1205971199101
120597119871
12059711990921198891199101)+119884
3(minus
1205972119871
120597119910112059711990931198891199101
+ 2119890minus2119891120597119891
1205971199101
120597
12059711990931198891199101)minus
120597119871
12059711991011198891199101
+1198831(119890
2119891 cos (1199093)1205972119871
120597119910212059711991011198891199102
+ 21198902119891 cos (1199093)120597119891
1205971199102
120597119871
12059711991011198891199102 + 119890
2119891 sin (1199093)1205972119871
120597119910221198891199102
+ 21198902119891 sin (1199093)120597119891
1205971199102
120597119871
12059711991021198891199102)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119910212059711991011198891199102
minus 21198902119891 sin (1199093)120597119891
1205971199102
120597119871
12059711991011198891199102 + 119890
2119891 cos (1199093)1205972119871
120597119910221198891199102
+ 21198902119891 cos (1199093)120597119891
1205971199102
120597119871
12059711991021198891199102)
+1198833(119890
2119891 1205972119871
120597119910212059711991031198891199102 + 2119890
2119891 120597119891
1205971199102
120597119871
12059711991031198891199102)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119910212059711990911198891199102
+ 2119890minus2119891 cos (1199093)120597119891
1205971199102
120597119871
12059711990911198891199102
+ 119890minus2119891 sin (1199093)
1205972119871
120597119910212059711990921198891199102
minus 2119890minus2119891 sin (1199093)120597119891
1205971199102
120597119871
12059711990921198891199102)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119910212059711990911198891199102
+ 2119890minus2119891 sin (1199093)120597119891
1205971199102
120597119871
12059711990911198891199102
minus 119890minus2119891 cos (1199093)
1205972119871
120597119910212059711990921198891199102
+ 2119890minus2119891 cos (1199093)120597119891
1205971199102
120597119871
12059711990921198891199102)+119884
3(minus
1205972119871
120597119910212059711990931198891199102
+ 2119890minus2119891120597119891
1205971199102
120597
12059711990931198891199102)minus
120597119871
12059711991021198891199102
+1198831(119890
2119891 cos (1199093)1205972119871
120597119910312059711991011198891199103
+ 21198902119891 cos (1199093)120597119891
1205971199103
120597119871
12059711991011198891199103
+ 1198902119891 sin (1199093)
1205972119871
120597119910312059711991021198891199103
+ 21198902119891 sin (1199093)120597119891
1205971199103
120597119871
12059711991021198891199103)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119910312059711991011198891199103
minus 21198902119891 sin (1199093)120597119891
1205971199103
120597119871
12059711991011198891199103
8 Advances in Mathematical Physics
+ 1198902119891 cos (1199093)
1205972119871
120597119910312059711991021198891199103
+ 21198902119891 cos (1199093)120597119891
1205971199103
120597119871
12059711991021198891199103)+119883
3(119890
2119891 1205972119871
120597119910231198891199103
+ 21198902119891120597119891
1205971199103
120597119871
12059711991031198891199103)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119910312059711990911198891199103
+ 2119890minus2119891 cos (1199093)120597119891
1205971199103
120597119871
12059711990911198891199103
+ 119890minus2119891 sin (1199093)
1205972119871
120597119910312059711990921198891199103
minus 2119890minus2119891 sin (1199093)120597119891
1205971199103
120597119871
12059711990921198891199103)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119910312059711990911198891199103
+ 2119890minus2119891 sin (1199093)120597119891
1205971199103
120597119871
12059711990911198891199103
minus 119890minus2119891 cos (1199093)
1205972119871
120597119910312059711990921198891199103
+ 2119890minus2119891 cos (1199093)120597119891
1205971199103
120597119871
12059711990921198891199103)+119884
3(minus
1205972119871
120597119910312059711990931198891199103
+ 2119890minus2119891120597119891
1205971199103
120597
12059711990931198891199103)minus
120597119871
12059711991031198891199103
(18)
Using (11) we get first equations as follows
1198831[minus119890
2119891 cos (1199093)1205972119871
120597119909112059711991011198891199091
minus 11989021198912
120597119891
1205971199091cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119909112059711991021198891199091
minus 11989021198912
120597119891
1205971199091sin (1199093)
120597119871
12059711991021198891199091]
+1198832[minus119890
2119891 cos (1199093)1205972119871
120597119909212059711991011198891199091
minus 11989021198912
120597119891
1205971199092cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119909212059711991021198891199091
minus 11989021198912
120597119891
1205971199092sin (1199093)
120597119871
12059711991021198891199091]
+1198833[minus119890
2119891 cos (1199093)1205972119871
120597119909312059711991011198891199091
minus 11989021198912
120597119891
1205971199093cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119909312059711991021198891199091
minus 11989021198912
120597119891
1205971199093sin (1199093)
120597119871
12059711991021198891199091]
+1198841[minus119890
2119891 cos (1199093)1205972119871
120597119910211198891199091
minus 11989021198912
120597119891
1205971199101cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119910112059711991021198891199091
minus 11989021198912
120597119891
1205971199101sin (1199093)
120597119871
12059711991021198891199091]
+1198842[minus119890
2119891 cos (1199093)1205972119871
120597119910212059711991011198891199091
minus 11989021198912
120597119891
1205971199102cos (1199093)
120597119871
12059711991011198891199091 minus 119890
2119891 sin (1199093)1205972119871
120597119910221198891199091
minus 11989021198912
120597119891
1205971199102sin (1199093)
120597119871
12059711991021198891199091]
+1198843[minus119890
2119891 cos (1199093)1205972119871
120597119910312059711991011198891199091
minus 11989021198912
120597119891
1205971199103cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119910312059711991021198891199091
minus 11989021198912
120597119891
1205971199103sin (1199093)
120597119871
12059711991021198891199091] = minus
120597119871
12059711990911198891199091
(19)
From here
minus cos (1199093) 119881(1198902119891 120597119871
1205971199101)minus sin (1199093) 119881(119890
2119891 120597119871
1205971199102)+
120597119871
1205971199091
= 0
(20)
Advances in Mathematical Physics 9
If we think of the curve 120572 for all equations as an integralcurve of 119881 that is 119881(120572) = (120597120597119905)(120572) we find the followingequations
(PDE1) minus cos (1199093)120597
120597119905(119890
2119891 120597119871
1205971199101)
minus sin (1199093)120597
120597119905(119890
2119891 120597119871
1205971199102)+
120597119871
1205971199091= 0
(PDE2) sin (1199093)120597
120597119905(119890
2119891 120597119871
1205971199101)
minus cos (1199093)120597
120597119905(119890
2119891 120597119871
1205971199102)+
120597119871
1205971199092= 0
(PDE3) minus120597
120597119905(119890
2119891 120597119871
1205971199103)+
120597119871
1205971199093= 0
(PDE4) cos (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199091)
minus sin (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199092)+
120597119871
1205971199101= 0
(PDE5) sin (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199091)
+ cos (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199092)+
120597119871
1205971199102= 0
(PDE6) 120597
120597119905(119890minus2119891 120597119871
1205971199093)+
120597119871
1205971199103= 0
(21)
such that the differential equations (21) are named conformalEuler-Lagrange equations on conformally flat manifold whichis shown in the form of (119879119872 119892 nabla 119869) Also therefore the triple(119879119872Φ
119871 119881) is called a conformal-Lagrangian mechanical
system on (119879119872 119892 nabla 119869)
7 Weyl-Euler-Lagrangian Equations forConservative Dynamical Systems
Proposition 26 We choose 119865 = i119881 119892 = Φ
119871 and 120582 = 2119891
at (11) and by considering (4) we can write Weyl-Lagrangiandynamic equation as follows
i119881(1198902119891
Φ119871) = i119881(Φ119871) minus 119889 (2119891) (22)
The second part (11) according to the law of conservation ofenergy [32] will not change for conservative dynamical systemsand i119881(Φ119871) = Φ
119871(119881)
Φ119871(119881) minus 2119889119891 = 119889119864
119871
Φ119871(119881) = 119889119864
119871+ 2119889119891 = 119889 (119864
119871+ 2119891)
(23)
From (21) above 119871 rarr 119871 + 2119891 So we can write
(PDE7) minus cos (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199101
)
minus sin (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199102
)
+120597 (119871 + 2119891)
1205971199091= 0
(PDE8) sin (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199101
)
minus cos (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199102
)
+120597 (119871 + 2119891)
1205971199092= 0
(PDE9) minus120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199103
)+120597 (119871 + 2119891)
1205971199093= 0
(PDE10) cos (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199091)
minus sin (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199092)
+120597 (119871 + 2119891)
1205971199101= 0
(PDE11) sin (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199091)
+ cos (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199092)
+120597 (119871 + 2119891)
1205971199102= 0
(PDE12) 120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199093)+
120597 (119871 + 2119891)1205971199103
= 0
(24)
and these differential equations (24) are named Weyl-Euler-Lagrange equations for conservative dynamical systems whichare constructed on conformally flat manifold (119879119872 119892 nabla 119869 119865)
and therefore the triple (119879119872Φ119871 119881) is called a Weyl-
Lagrangian mechanical system
8 Equations Solving with Computer
Theequations systems (21) and (24) have been solved by usingthe symbolic Algebra software and implicit solution is below
119871 (1199091 1199092 1199093 1199101 1199102 1199103 119905)
= exp (minus119894 lowast 119905) lowast 1198651 (1199103 minus 119894 lowast 1199093) + 1198652 (119905)
+ exp (119905 lowast 119894) lowast 1198653 (1199103 +1199093 lowast 119894) for 119891 = 0
(25)
10 Advances in Mathematical Physics
05
1
05 1 15 2minus1
minus1
minus05
minus05
(a)
05
1
05 1 15 2minus1
minus1
minus05
minus05
(b)
Figure 1
It is well known that an electromagnetic field is a physical fieldproduced by electrically charged objects The movement ofobjects in electrical magnetic and gravitational fields force isvery important For instance on a weather map the surfacewind velocity is defined by assigning a vector to each pointon a map So each vector represents the speed and directionof the movement of air at that point
The location of each object in space is represented bythree dimensions in physical space The dimensions whichare represented by higher dimensions are time positionmass and so forth The number of dimensions of (25) will bereduced to three and behind the graphics will be drawn Firstimplicit function at (25) will be selected as special After thefigure of (25) has been drawn for the route of the movementof objects in the electromagnetic field
Example 27 Consider
119871 (1199091 1199092 1199093 1199101 1199102 1199103 119905) = exp (minus119894 lowast 119905) + exp (119905 lowast 119894) lowast 119905 minus 1199052 (26)
(see Figure 1)
9 Discussion
A classical field theory explains the study of how one or morephysical fields interact with matter which is used in quantumand classical mechanics of physics branches In this study theEuler-Lagrange mechanical equations (21) and (24) derivedon a generalized on flat manifolds may be suggested to dealwith problems in electrical magnetic and gravitational fieldsforce for the path of movement (26) of defined space movingobjects [24]
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the agency BAP of PamukkaleUniversity
References
[1] W K Liu and S Jun Computational Nanomechanics of Mate-rials American Scientific Publishers Stevenson Ranch CalifUSA 2005
[2] M Tekkoyun and Y Yayli ldquoMechanical systems on generalized-quaternionic Kahler manifoldsrdquo International Journal of Geo-metric Methods in Modern Physics vol 8 no 7 pp 1419ndash14312011
[3] D B Fogel Epistemology of a theory of everything Weyl Ein-stein and the unification of physics [PhD thesis] GraduateSchool of theUniversity ofNotreDameNotreDame Ind USA2008
[4] Z Kasap and M Tekkoyun ldquoMechanical systems on almostparapseudo-KahlerndashWeyl manifoldsrdquo International Journal ofGeometric Methods in Modern Physics vol 10 no 5 Article ID1350008 8 pages 2013
[5] Z Kasap ldquoWeyl-mechanical systems on tangent manifoldsof constant 119882-sectional curvaturerdquo International Journal ofGeometric Methods in Modern Physics vol 10 no 10 Article ID1350053 pp 1ndash13 2013
[6] M Kapovich ldquoFlat conformal structures on 3-manifolds I uni-formization of closed seifert manifoldsrdquo Journal of DifferentialGeometry vol 38 no 1 pp 191ndash215 1993
[7] F Schwartz ldquoA volumetric Penrose inequality for conformallyflat manifoldsrdquo Annales Henri Poincare vol 12 no 1 pp 67ndash762011
[8] R S Kulkarni ldquoConformally flat manifoldsrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 69 pp 2675ndash2676 1972
[9] I G Dotti and R J Miatello ldquoOn the cohomology ring offlat manifolds with a special structurerdquo Revista De La Uni OnMatematica Argentina vol 46 no 2 pp 133ndash147 2005
Advances in Mathematical Physics 11
[10] A Szczepanski Kahler at Manifolds of Low Dimensions InstitutdesHautes Etudes Scientifiques Bures-sur-Yvette France 2005
[11] R Bartnik ldquoThemass of an asymptotically flat manifoldrdquo Com-munications on Pure and AppliedMathematics vol 39 no 5 pp661ndash693 1986
[12] M D M Gonzalez ldquoSingular sets of a class of locally confor-mally flat manifoldsrdquo Duke Mathematical Journal vol 129 no3 pp 551ndash572 2005
[13] S Akbulut and M Kalafat ldquoA class of locally conformally flat4-manifoldsrdquoNewYork Journal of Mathematics vol 18 pp 733ndash763 2012
[14] S-H Zhu ldquoThe classification of complete locally conformallyflat manifolds of nonnegative Ricci curvaturerdquo Pacific Journalof Mathematics vol 163 no 1 pp 189ndash199 1994
[15] H M Abood ldquoAlmost Hermitian manifold with flat Bochnertensorrdquo European Journal of Pure and Applied Mathematics vol3 no 4 pp 730ndash736 2010
[16] K Olszak and Z Olszak ldquoOn 4-dimensional conformally flatalmost 120576-Kahlerianmanifoldsrdquo Journal of Geometry and Physicsvol 62 no 5 pp 1108ndash1113 2012
[17] S Upadhyay ldquoA bounding question for almost flat manifoldsrdquoTransactions of the AmericanMathematical Society vol 353 no3 pp 963ndash972 2001
[18] 2015 httpmathworldwolframcomFlatManifoldhtml[19] 2015 httpenwikipediaorgwikiConformally flat manifold[20] D McDu and D Salamon J-Holomorphic Curves and Quantum
Cohomology AMS 1995[21] 2015 httpenwikipediaorgwikiConformal class[22] G B Folland ldquoWeyl manifoldsrdquo Journal of Differential Geome-
try vol 4 pp 145ndash153 1970[23] L Kadosh Topics in weyl geometry [PhD thesis] University of
California Berkeley Calif USA 1996[24] H Weyl Space-Time-Matter Dover Publications 1922 Trans-
lated from the 4th German edition by H Brose MethuenLondon UK Dover Publications New York NY USA 1952
[25] P Gilkey and S Nikcevic ldquoKahler and para-Kahler curvatureWeyl manifoldsrdquo httparxivorgabs10114844
[26] H Pedersen Y S Poon and A Swann ldquoThe Einstein-Weylequations in complex and quaternionic geometryrdquo DifferentialGeometry and Its Applications vol 3 no 4 pp 309ndash321 1993
[27] P Gilkey and S Nikcevic ldquoKahler-Weylmanifolds of dimension4rdquo httparxivorgabs11094532
[28] M Brozos-Vazquez P Gilkey and E Merino ldquoGeometricrealizations of Kaehler and of para-Kaehler curvature modelsrdquoInternational Journal of Geometric Methods in Modern Physicsvol 7 no 3 pp 505ndash515 2010
[29] 2015 httpenwikipediaorgwikiWeyl tensor[30] R Miron D Hrimiuc H Shimada and S V SabauThe Geom-
etry of Hamilton and Lagrange Spaces Kluwer Academic Pub-lishers 2002
[31] J Klein ldquoEspaces variationnels et mecaniquerdquo Annales delrsquoInstitut Fourier vol 12 pp 1ndash124 1962
[32] M de Leon and P R RodriguesMethods of Differential Geom-etry in Analytical Mechanics North-Holland Elsevier Amster-dam The Netherlands 1989
[33] R Abraham J E Marsden and T Ratiu Manifolds TensorAnalysis and Applications Springer New York NY USA 2001
Submit your manuscripts athttpwwwhindawicom
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MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Journal of
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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International Journal of
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Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Advances in Mathematical Physics
+ 21198902119891 cos (1199093)120597119891
1205971199092
120597119871
12059711991011198891199092
+ 1198902119891 sin (1199093)
1205972119871
120597119909212059711991021198891199092
+ 21198902119891 sin (1199093)120597119891
1205971199092
120597119871
12059711991021198891199092)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119909212059711991011198891199092
minus 21198902119891 sin (1199093)120597119891
1205971199092
120597119871
12059711991011198891199092
+ 1198902119891 cos (1199093)
1205972119871
120597119909212059711991021198891199092
+ 21198902119891 cos (1199093)120597119891
1205971199092
120597119871
12059711991021198891199092)
+1198833(119890
2119891 1205972119871
120597119909212059711991031198891199092 + 2119890
2119891 120597119891
1205971199092
120597119871
12059711991031198891199092)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119909212059711990911198891199092
+ 2119890minus2119891 cos (1199093)120597119891
1205971199092
120597119871
12059711990911198891199092
+ 119890minus2119891 sin (1199093)
1205972119871
120597119909221198891199092
minus 2119890minus2119891 sin (1199093)120597119891
1205971199092
120597119871
12059711990921198891199092)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119909212059711990911198891199092
+ 2119890minus2119891 sin (1199093)120597119891
1205971199092
120597119871
12059711990911198891199092
minus 119890minus2119891 cos (1199093)
1205972119871
120597119909221198891199092
+ 2119890minus2119891 cos (1199093)120597119891
1205971199092
120597119871
12059711990921198891199092)
+1198843(minus
1205972119871
120597119909212059711990931198891199092 + 2119890
minus2119891 120597119891
1205971199092
120597
12059711990931198891199092)
minus120597119871
12059711990921198891199092 +119883
1(119890
2119891 cos (1199093)1205972119871
120597119909312059711991011198891199093
+ 21198902119891 cos (1199093)120597119891
1205971199093
120597119871
12059711991011198891199093 minus 119890
2119891 sin (1199093)120597119871
12059711991011198891199093
+ 1198902119891 sin (1199093)
1205972119871
120597119909312059711991021198891199093
+ 21198902119891 sin (1199093)120597119891
1205971199093
120597119871
12059711991021198891199093
+ 1198902119891 cos (1199093)
120597119871
12059711991021198891199093)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119909312059711991011198891199093
minus 21198902119891 sin (1199093)120597119891
1205971199093
120597119871
12059711991011198891199093 minus 119890
2119891 cos (1199093)120597119871
12059711991011198891199093
+ 1198902119891 cos (1199093)
1205972119871
120597119909312059711991021198891199093
+ 21198902119891 cos (1199093)120597119891
1205971199093
120597119871
12059711991021198891199093
minus 1198902119891 sin (1199093)
120597119871
12059711991021198891199093)+119883
3(119890
2119891 1205972119871
120597119909312059711991031198891199093
+ 21198902119891120597119891
1205971199093
120597119871
12059711991031198891199093)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119909312059711990911198891199093
+ 2119890minus2119891 cos (1199093)120597119891
1205971199093
120597119871
12059711990911198891199093
minus 119890minus2119891 sin (1199093)
120597119871
12059711990911198891199093 + 119890
minus2119891 sin (1199093)1205972119871
120597119909312059711990921198891199093
minus 2119890minus2119891 sin (1199093)120597119891
1205971199093
120597119871
12059711990921198891199093
+ 119890minus2119891 cos (1199093)
120597119871
12059711990921198891199093)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119909312059711990911198891199093
+ 2119890minus2119891 sin (1199093)120597119891
1205971199093
120597119871
12059711990911198891199093
minus 119890minus2119891 cos (1199093)
120597119871
12059711990911198891199093
minus 119890minus2119891 cos (1199093)
1205972119871
120597119909312059711990921198891199093
+ 2119890minus2119891 cos (1199093)120597119891
1205971199093
120597119871
12059711990921198891199093
+ 119890minus2119891 sin (1199093)
120597119871
12059711990921198891199093)+119884
3(minus
1205972119871
120597119909231198891199093
+ 2119890minus2119891120597119891
1205971199093
120597
12059711990931198891199093)minus
120597119871
12059711990931198891199093
Advances in Mathematical Physics 7
+1198831(119890
2119891 cos (1199093)1205972119871
120597119910211198891199101
+ 21198902119891 cos (1199093)120597119891
1205971199101
120597119871
12059711991011198891199101
+ 1198902119891 sin (1199093)
1205972119871
120597119910112059711991021198891199101
+ 21198902119891 sin (1199093)120597119891
1205971199101
120597119871
12059711991021198891199101)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119910211198891199101
minus 21198902119891 sin (1199093)120597119891
1205971199101
120597119871
12059711991011198891199101
+ 1198902119891 cos (1199093)
1205972119871
120597119910112059711991021198891199101
+ 21198902119891 cos (1199093)120597119891
1199101
120597119871
12059711991021198891199101)+119883
3(119890
2119891 1205972119871
120597119910112059711991031198891199101
+ 21198902119891120597119891
1205971199101
120597119871
12059711991031198891199101)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119910112059711990911198891199101
+ 2119890minus2119891 cos (1199093)120597119891
1205971199101
120597119871
12059711990911198891199101
+ 119890minus2119891 sin (1199093)
1205972119871
120597119910112059711990921198891199101
minus 2119890minus2119891 sin (1199093)120597119891
1205971199101
120597119871
12059711990921198891199101)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119910112059711990911198891199101
+ 2119890minus2119891 sin (1199093)120597119891
1205971199101
120597119871
12059711990911198891199101
minus 119890minus2119891 cos (1199093)
1205972119871
120597119910112059711990921198891199101
+ 2119890minus2119891 cos (1199093)120597119891
1205971199101
120597119871
12059711990921198891199101)+119884
3(minus
1205972119871
120597119910112059711990931198891199101
+ 2119890minus2119891120597119891
1205971199101
120597
12059711990931198891199101)minus
120597119871
12059711991011198891199101
+1198831(119890
2119891 cos (1199093)1205972119871
120597119910212059711991011198891199102
+ 21198902119891 cos (1199093)120597119891
1205971199102
120597119871
12059711991011198891199102 + 119890
2119891 sin (1199093)1205972119871
120597119910221198891199102
+ 21198902119891 sin (1199093)120597119891
1205971199102
120597119871
12059711991021198891199102)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119910212059711991011198891199102
minus 21198902119891 sin (1199093)120597119891
1205971199102
120597119871
12059711991011198891199102 + 119890
2119891 cos (1199093)1205972119871
120597119910221198891199102
+ 21198902119891 cos (1199093)120597119891
1205971199102
120597119871
12059711991021198891199102)
+1198833(119890
2119891 1205972119871
120597119910212059711991031198891199102 + 2119890
2119891 120597119891
1205971199102
120597119871
12059711991031198891199102)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119910212059711990911198891199102
+ 2119890minus2119891 cos (1199093)120597119891
1205971199102
120597119871
12059711990911198891199102
+ 119890minus2119891 sin (1199093)
1205972119871
120597119910212059711990921198891199102
minus 2119890minus2119891 sin (1199093)120597119891
1205971199102
120597119871
12059711990921198891199102)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119910212059711990911198891199102
+ 2119890minus2119891 sin (1199093)120597119891
1205971199102
120597119871
12059711990911198891199102
minus 119890minus2119891 cos (1199093)
1205972119871
120597119910212059711990921198891199102
+ 2119890minus2119891 cos (1199093)120597119891
1205971199102
120597119871
12059711990921198891199102)+119884
3(minus
1205972119871
120597119910212059711990931198891199102
+ 2119890minus2119891120597119891
1205971199102
120597
12059711990931198891199102)minus
120597119871
12059711991021198891199102
+1198831(119890
2119891 cos (1199093)1205972119871
120597119910312059711991011198891199103
+ 21198902119891 cos (1199093)120597119891
1205971199103
120597119871
12059711991011198891199103
+ 1198902119891 sin (1199093)
1205972119871
120597119910312059711991021198891199103
+ 21198902119891 sin (1199093)120597119891
1205971199103
120597119871
12059711991021198891199103)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119910312059711991011198891199103
minus 21198902119891 sin (1199093)120597119891
1205971199103
120597119871
12059711991011198891199103
8 Advances in Mathematical Physics
+ 1198902119891 cos (1199093)
1205972119871
120597119910312059711991021198891199103
+ 21198902119891 cos (1199093)120597119891
1205971199103
120597119871
12059711991021198891199103)+119883
3(119890
2119891 1205972119871
120597119910231198891199103
+ 21198902119891120597119891
1205971199103
120597119871
12059711991031198891199103)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119910312059711990911198891199103
+ 2119890minus2119891 cos (1199093)120597119891
1205971199103
120597119871
12059711990911198891199103
+ 119890minus2119891 sin (1199093)
1205972119871
120597119910312059711990921198891199103
minus 2119890minus2119891 sin (1199093)120597119891
1205971199103
120597119871
12059711990921198891199103)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119910312059711990911198891199103
+ 2119890minus2119891 sin (1199093)120597119891
1205971199103
120597119871
12059711990911198891199103
minus 119890minus2119891 cos (1199093)
1205972119871
120597119910312059711990921198891199103
+ 2119890minus2119891 cos (1199093)120597119891
1205971199103
120597119871
12059711990921198891199103)+119884
3(minus
1205972119871
120597119910312059711990931198891199103
+ 2119890minus2119891120597119891
1205971199103
120597
12059711990931198891199103)minus
120597119871
12059711991031198891199103
(18)
Using (11) we get first equations as follows
1198831[minus119890
2119891 cos (1199093)1205972119871
120597119909112059711991011198891199091
minus 11989021198912
120597119891
1205971199091cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119909112059711991021198891199091
minus 11989021198912
120597119891
1205971199091sin (1199093)
120597119871
12059711991021198891199091]
+1198832[minus119890
2119891 cos (1199093)1205972119871
120597119909212059711991011198891199091
minus 11989021198912
120597119891
1205971199092cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119909212059711991021198891199091
minus 11989021198912
120597119891
1205971199092sin (1199093)
120597119871
12059711991021198891199091]
+1198833[minus119890
2119891 cos (1199093)1205972119871
120597119909312059711991011198891199091
minus 11989021198912
120597119891
1205971199093cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119909312059711991021198891199091
minus 11989021198912
120597119891
1205971199093sin (1199093)
120597119871
12059711991021198891199091]
+1198841[minus119890
2119891 cos (1199093)1205972119871
120597119910211198891199091
minus 11989021198912
120597119891
1205971199101cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119910112059711991021198891199091
minus 11989021198912
120597119891
1205971199101sin (1199093)
120597119871
12059711991021198891199091]
+1198842[minus119890
2119891 cos (1199093)1205972119871
120597119910212059711991011198891199091
minus 11989021198912
120597119891
1205971199102cos (1199093)
120597119871
12059711991011198891199091 minus 119890
2119891 sin (1199093)1205972119871
120597119910221198891199091
minus 11989021198912
120597119891
1205971199102sin (1199093)
120597119871
12059711991021198891199091]
+1198843[minus119890
2119891 cos (1199093)1205972119871
120597119910312059711991011198891199091
minus 11989021198912
120597119891
1205971199103cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119910312059711991021198891199091
minus 11989021198912
120597119891
1205971199103sin (1199093)
120597119871
12059711991021198891199091] = minus
120597119871
12059711990911198891199091
(19)
From here
minus cos (1199093) 119881(1198902119891 120597119871
1205971199101)minus sin (1199093) 119881(119890
2119891 120597119871
1205971199102)+
120597119871
1205971199091
= 0
(20)
Advances in Mathematical Physics 9
If we think of the curve 120572 for all equations as an integralcurve of 119881 that is 119881(120572) = (120597120597119905)(120572) we find the followingequations
(PDE1) minus cos (1199093)120597
120597119905(119890
2119891 120597119871
1205971199101)
minus sin (1199093)120597
120597119905(119890
2119891 120597119871
1205971199102)+
120597119871
1205971199091= 0
(PDE2) sin (1199093)120597
120597119905(119890
2119891 120597119871
1205971199101)
minus cos (1199093)120597
120597119905(119890
2119891 120597119871
1205971199102)+
120597119871
1205971199092= 0
(PDE3) minus120597
120597119905(119890
2119891 120597119871
1205971199103)+
120597119871
1205971199093= 0
(PDE4) cos (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199091)
minus sin (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199092)+
120597119871
1205971199101= 0
(PDE5) sin (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199091)
+ cos (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199092)+
120597119871
1205971199102= 0
(PDE6) 120597
120597119905(119890minus2119891 120597119871
1205971199093)+
120597119871
1205971199103= 0
(21)
such that the differential equations (21) are named conformalEuler-Lagrange equations on conformally flat manifold whichis shown in the form of (119879119872 119892 nabla 119869) Also therefore the triple(119879119872Φ
119871 119881) is called a conformal-Lagrangian mechanical
system on (119879119872 119892 nabla 119869)
7 Weyl-Euler-Lagrangian Equations forConservative Dynamical Systems
Proposition 26 We choose 119865 = i119881 119892 = Φ
119871 and 120582 = 2119891
at (11) and by considering (4) we can write Weyl-Lagrangiandynamic equation as follows
i119881(1198902119891
Φ119871) = i119881(Φ119871) minus 119889 (2119891) (22)
The second part (11) according to the law of conservation ofenergy [32] will not change for conservative dynamical systemsand i119881(Φ119871) = Φ
119871(119881)
Φ119871(119881) minus 2119889119891 = 119889119864
119871
Φ119871(119881) = 119889119864
119871+ 2119889119891 = 119889 (119864
119871+ 2119891)
(23)
From (21) above 119871 rarr 119871 + 2119891 So we can write
(PDE7) minus cos (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199101
)
minus sin (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199102
)
+120597 (119871 + 2119891)
1205971199091= 0
(PDE8) sin (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199101
)
minus cos (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199102
)
+120597 (119871 + 2119891)
1205971199092= 0
(PDE9) minus120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199103
)+120597 (119871 + 2119891)
1205971199093= 0
(PDE10) cos (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199091)
minus sin (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199092)
+120597 (119871 + 2119891)
1205971199101= 0
(PDE11) sin (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199091)
+ cos (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199092)
+120597 (119871 + 2119891)
1205971199102= 0
(PDE12) 120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199093)+
120597 (119871 + 2119891)1205971199103
= 0
(24)
and these differential equations (24) are named Weyl-Euler-Lagrange equations for conservative dynamical systems whichare constructed on conformally flat manifold (119879119872 119892 nabla 119869 119865)
and therefore the triple (119879119872Φ119871 119881) is called a Weyl-
Lagrangian mechanical system
8 Equations Solving with Computer
Theequations systems (21) and (24) have been solved by usingthe symbolic Algebra software and implicit solution is below
119871 (1199091 1199092 1199093 1199101 1199102 1199103 119905)
= exp (minus119894 lowast 119905) lowast 1198651 (1199103 minus 119894 lowast 1199093) + 1198652 (119905)
+ exp (119905 lowast 119894) lowast 1198653 (1199103 +1199093 lowast 119894) for 119891 = 0
(25)
10 Advances in Mathematical Physics
05
1
05 1 15 2minus1
minus1
minus05
minus05
(a)
05
1
05 1 15 2minus1
minus1
minus05
minus05
(b)
Figure 1
It is well known that an electromagnetic field is a physical fieldproduced by electrically charged objects The movement ofobjects in electrical magnetic and gravitational fields force isvery important For instance on a weather map the surfacewind velocity is defined by assigning a vector to each pointon a map So each vector represents the speed and directionof the movement of air at that point
The location of each object in space is represented bythree dimensions in physical space The dimensions whichare represented by higher dimensions are time positionmass and so forth The number of dimensions of (25) will bereduced to three and behind the graphics will be drawn Firstimplicit function at (25) will be selected as special After thefigure of (25) has been drawn for the route of the movementof objects in the electromagnetic field
Example 27 Consider
119871 (1199091 1199092 1199093 1199101 1199102 1199103 119905) = exp (minus119894 lowast 119905) + exp (119905 lowast 119894) lowast 119905 minus 1199052 (26)
(see Figure 1)
9 Discussion
A classical field theory explains the study of how one or morephysical fields interact with matter which is used in quantumand classical mechanics of physics branches In this study theEuler-Lagrange mechanical equations (21) and (24) derivedon a generalized on flat manifolds may be suggested to dealwith problems in electrical magnetic and gravitational fieldsforce for the path of movement (26) of defined space movingobjects [24]
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the agency BAP of PamukkaleUniversity
References
[1] W K Liu and S Jun Computational Nanomechanics of Mate-rials American Scientific Publishers Stevenson Ranch CalifUSA 2005
[2] M Tekkoyun and Y Yayli ldquoMechanical systems on generalized-quaternionic Kahler manifoldsrdquo International Journal of Geo-metric Methods in Modern Physics vol 8 no 7 pp 1419ndash14312011
[3] D B Fogel Epistemology of a theory of everything Weyl Ein-stein and the unification of physics [PhD thesis] GraduateSchool of theUniversity ofNotreDameNotreDame Ind USA2008
[4] Z Kasap and M Tekkoyun ldquoMechanical systems on almostparapseudo-KahlerndashWeyl manifoldsrdquo International Journal ofGeometric Methods in Modern Physics vol 10 no 5 Article ID1350008 8 pages 2013
[5] Z Kasap ldquoWeyl-mechanical systems on tangent manifoldsof constant 119882-sectional curvaturerdquo International Journal ofGeometric Methods in Modern Physics vol 10 no 10 Article ID1350053 pp 1ndash13 2013
[6] M Kapovich ldquoFlat conformal structures on 3-manifolds I uni-formization of closed seifert manifoldsrdquo Journal of DifferentialGeometry vol 38 no 1 pp 191ndash215 1993
[7] F Schwartz ldquoA volumetric Penrose inequality for conformallyflat manifoldsrdquo Annales Henri Poincare vol 12 no 1 pp 67ndash762011
[8] R S Kulkarni ldquoConformally flat manifoldsrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 69 pp 2675ndash2676 1972
[9] I G Dotti and R J Miatello ldquoOn the cohomology ring offlat manifolds with a special structurerdquo Revista De La Uni OnMatematica Argentina vol 46 no 2 pp 133ndash147 2005
Advances in Mathematical Physics 11
[10] A Szczepanski Kahler at Manifolds of Low Dimensions InstitutdesHautes Etudes Scientifiques Bures-sur-Yvette France 2005
[11] R Bartnik ldquoThemass of an asymptotically flat manifoldrdquo Com-munications on Pure and AppliedMathematics vol 39 no 5 pp661ndash693 1986
[12] M D M Gonzalez ldquoSingular sets of a class of locally confor-mally flat manifoldsrdquo Duke Mathematical Journal vol 129 no3 pp 551ndash572 2005
[13] S Akbulut and M Kalafat ldquoA class of locally conformally flat4-manifoldsrdquoNewYork Journal of Mathematics vol 18 pp 733ndash763 2012
[14] S-H Zhu ldquoThe classification of complete locally conformallyflat manifolds of nonnegative Ricci curvaturerdquo Pacific Journalof Mathematics vol 163 no 1 pp 189ndash199 1994
[15] H M Abood ldquoAlmost Hermitian manifold with flat Bochnertensorrdquo European Journal of Pure and Applied Mathematics vol3 no 4 pp 730ndash736 2010
[16] K Olszak and Z Olszak ldquoOn 4-dimensional conformally flatalmost 120576-Kahlerianmanifoldsrdquo Journal of Geometry and Physicsvol 62 no 5 pp 1108ndash1113 2012
[17] S Upadhyay ldquoA bounding question for almost flat manifoldsrdquoTransactions of the AmericanMathematical Society vol 353 no3 pp 963ndash972 2001
[18] 2015 httpmathworldwolframcomFlatManifoldhtml[19] 2015 httpenwikipediaorgwikiConformally flat manifold[20] D McDu and D Salamon J-Holomorphic Curves and Quantum
Cohomology AMS 1995[21] 2015 httpenwikipediaorgwikiConformal class[22] G B Folland ldquoWeyl manifoldsrdquo Journal of Differential Geome-
try vol 4 pp 145ndash153 1970[23] L Kadosh Topics in weyl geometry [PhD thesis] University of
California Berkeley Calif USA 1996[24] H Weyl Space-Time-Matter Dover Publications 1922 Trans-
lated from the 4th German edition by H Brose MethuenLondon UK Dover Publications New York NY USA 1952
[25] P Gilkey and S Nikcevic ldquoKahler and para-Kahler curvatureWeyl manifoldsrdquo httparxivorgabs10114844
[26] H Pedersen Y S Poon and A Swann ldquoThe Einstein-Weylequations in complex and quaternionic geometryrdquo DifferentialGeometry and Its Applications vol 3 no 4 pp 309ndash321 1993
[27] P Gilkey and S Nikcevic ldquoKahler-Weylmanifolds of dimension4rdquo httparxivorgabs11094532
[28] M Brozos-Vazquez P Gilkey and E Merino ldquoGeometricrealizations of Kaehler and of para-Kaehler curvature modelsrdquoInternational Journal of Geometric Methods in Modern Physicsvol 7 no 3 pp 505ndash515 2010
[29] 2015 httpenwikipediaorgwikiWeyl tensor[30] R Miron D Hrimiuc H Shimada and S V SabauThe Geom-
etry of Hamilton and Lagrange Spaces Kluwer Academic Pub-lishers 2002
[31] J Klein ldquoEspaces variationnels et mecaniquerdquo Annales delrsquoInstitut Fourier vol 12 pp 1ndash124 1962
[32] M de Leon and P R RodriguesMethods of Differential Geom-etry in Analytical Mechanics North-Holland Elsevier Amster-dam The Netherlands 1989
[33] R Abraham J E Marsden and T Ratiu Manifolds TensorAnalysis and Applications Springer New York NY USA 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 7
+1198831(119890
2119891 cos (1199093)1205972119871
120597119910211198891199101
+ 21198902119891 cos (1199093)120597119891
1205971199101
120597119871
12059711991011198891199101
+ 1198902119891 sin (1199093)
1205972119871
120597119910112059711991021198891199101
+ 21198902119891 sin (1199093)120597119891
1205971199101
120597119871
12059711991021198891199101)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119910211198891199101
minus 21198902119891 sin (1199093)120597119891
1205971199101
120597119871
12059711991011198891199101
+ 1198902119891 cos (1199093)
1205972119871
120597119910112059711991021198891199101
+ 21198902119891 cos (1199093)120597119891
1199101
120597119871
12059711991021198891199101)+119883
3(119890
2119891 1205972119871
120597119910112059711991031198891199101
+ 21198902119891120597119891
1205971199101
120597119871
12059711991031198891199101)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119910112059711990911198891199101
+ 2119890minus2119891 cos (1199093)120597119891
1205971199101
120597119871
12059711990911198891199101
+ 119890minus2119891 sin (1199093)
1205972119871
120597119910112059711990921198891199101
minus 2119890minus2119891 sin (1199093)120597119891
1205971199101
120597119871
12059711990921198891199101)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119910112059711990911198891199101
+ 2119890minus2119891 sin (1199093)120597119891
1205971199101
120597119871
12059711990911198891199101
minus 119890minus2119891 cos (1199093)
1205972119871
120597119910112059711990921198891199101
+ 2119890minus2119891 cos (1199093)120597119891
1205971199101
120597119871
12059711990921198891199101)+119884
3(minus
1205972119871
120597119910112059711990931198891199101
+ 2119890minus2119891120597119891
1205971199101
120597
12059711990931198891199101)minus
120597119871
12059711991011198891199101
+1198831(119890
2119891 cos (1199093)1205972119871
120597119910212059711991011198891199102
+ 21198902119891 cos (1199093)120597119891
1205971199102
120597119871
12059711991011198891199102 + 119890
2119891 sin (1199093)1205972119871
120597119910221198891199102
+ 21198902119891 sin (1199093)120597119891
1205971199102
120597119871
12059711991021198891199102)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119910212059711991011198891199102
minus 21198902119891 sin (1199093)120597119891
1205971199102
120597119871
12059711991011198891199102 + 119890
2119891 cos (1199093)1205972119871
120597119910221198891199102
+ 21198902119891 cos (1199093)120597119891
1205971199102
120597119871
12059711991021198891199102)
+1198833(119890
2119891 1205972119871
120597119910212059711991031198891199102 + 2119890
2119891 120597119891
1205971199102
120597119871
12059711991031198891199102)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119910212059711990911198891199102
+ 2119890minus2119891 cos (1199093)120597119891
1205971199102
120597119871
12059711990911198891199102
+ 119890minus2119891 sin (1199093)
1205972119871
120597119910212059711990921198891199102
minus 2119890minus2119891 sin (1199093)120597119891
1205971199102
120597119871
12059711990921198891199102)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119910212059711990911198891199102
+ 2119890minus2119891 sin (1199093)120597119891
1205971199102
120597119871
12059711990911198891199102
minus 119890minus2119891 cos (1199093)
1205972119871
120597119910212059711990921198891199102
+ 2119890minus2119891 cos (1199093)120597119891
1205971199102
120597119871
12059711990921198891199102)+119884
3(minus
1205972119871
120597119910212059711990931198891199102
+ 2119890minus2119891120597119891
1205971199102
120597
12059711990931198891199102)minus
120597119871
12059711991021198891199102
+1198831(119890
2119891 cos (1199093)1205972119871
120597119910312059711991011198891199103
+ 21198902119891 cos (1199093)120597119891
1205971199103
120597119871
12059711991011198891199103
+ 1198902119891 sin (1199093)
1205972119871
120597119910312059711991021198891199103
+ 21198902119891 sin (1199093)120597119891
1205971199103
120597119871
12059711991021198891199103)
+1198832(minus119890
2119891 sin (1199093)1205972119871
120597119910312059711991011198891199103
minus 21198902119891 sin (1199093)120597119891
1205971199103
120597119871
12059711991011198891199103
8 Advances in Mathematical Physics
+ 1198902119891 cos (1199093)
1205972119871
120597119910312059711991021198891199103
+ 21198902119891 cos (1199093)120597119891
1205971199103
120597119871
12059711991021198891199103)+119883
3(119890
2119891 1205972119871
120597119910231198891199103
+ 21198902119891120597119891
1205971199103
120597119871
12059711991031198891199103)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119910312059711990911198891199103
+ 2119890minus2119891 cos (1199093)120597119891
1205971199103
120597119871
12059711990911198891199103
+ 119890minus2119891 sin (1199093)
1205972119871
120597119910312059711990921198891199103
minus 2119890minus2119891 sin (1199093)120597119891
1205971199103
120597119871
12059711990921198891199103)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119910312059711990911198891199103
+ 2119890minus2119891 sin (1199093)120597119891
1205971199103
120597119871
12059711990911198891199103
minus 119890minus2119891 cos (1199093)
1205972119871
120597119910312059711990921198891199103
+ 2119890minus2119891 cos (1199093)120597119891
1205971199103
120597119871
12059711990921198891199103)+119884
3(minus
1205972119871
120597119910312059711990931198891199103
+ 2119890minus2119891120597119891
1205971199103
120597
12059711990931198891199103)minus
120597119871
12059711991031198891199103
(18)
Using (11) we get first equations as follows
1198831[minus119890
2119891 cos (1199093)1205972119871
120597119909112059711991011198891199091
minus 11989021198912
120597119891
1205971199091cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119909112059711991021198891199091
minus 11989021198912
120597119891
1205971199091sin (1199093)
120597119871
12059711991021198891199091]
+1198832[minus119890
2119891 cos (1199093)1205972119871
120597119909212059711991011198891199091
minus 11989021198912
120597119891
1205971199092cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119909212059711991021198891199091
minus 11989021198912
120597119891
1205971199092sin (1199093)
120597119871
12059711991021198891199091]
+1198833[minus119890
2119891 cos (1199093)1205972119871
120597119909312059711991011198891199091
minus 11989021198912
120597119891
1205971199093cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119909312059711991021198891199091
minus 11989021198912
120597119891
1205971199093sin (1199093)
120597119871
12059711991021198891199091]
+1198841[minus119890
2119891 cos (1199093)1205972119871
120597119910211198891199091
minus 11989021198912
120597119891
1205971199101cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119910112059711991021198891199091
minus 11989021198912
120597119891
1205971199101sin (1199093)
120597119871
12059711991021198891199091]
+1198842[minus119890
2119891 cos (1199093)1205972119871
120597119910212059711991011198891199091
minus 11989021198912
120597119891
1205971199102cos (1199093)
120597119871
12059711991011198891199091 minus 119890
2119891 sin (1199093)1205972119871
120597119910221198891199091
minus 11989021198912
120597119891
1205971199102sin (1199093)
120597119871
12059711991021198891199091]
+1198843[minus119890
2119891 cos (1199093)1205972119871
120597119910312059711991011198891199091
minus 11989021198912
120597119891
1205971199103cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119910312059711991021198891199091
minus 11989021198912
120597119891
1205971199103sin (1199093)
120597119871
12059711991021198891199091] = minus
120597119871
12059711990911198891199091
(19)
From here
minus cos (1199093) 119881(1198902119891 120597119871
1205971199101)minus sin (1199093) 119881(119890
2119891 120597119871
1205971199102)+
120597119871
1205971199091
= 0
(20)
Advances in Mathematical Physics 9
If we think of the curve 120572 for all equations as an integralcurve of 119881 that is 119881(120572) = (120597120597119905)(120572) we find the followingequations
(PDE1) minus cos (1199093)120597
120597119905(119890
2119891 120597119871
1205971199101)
minus sin (1199093)120597
120597119905(119890
2119891 120597119871
1205971199102)+
120597119871
1205971199091= 0
(PDE2) sin (1199093)120597
120597119905(119890
2119891 120597119871
1205971199101)
minus cos (1199093)120597
120597119905(119890
2119891 120597119871
1205971199102)+
120597119871
1205971199092= 0
(PDE3) minus120597
120597119905(119890
2119891 120597119871
1205971199103)+
120597119871
1205971199093= 0
(PDE4) cos (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199091)
minus sin (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199092)+
120597119871
1205971199101= 0
(PDE5) sin (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199091)
+ cos (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199092)+
120597119871
1205971199102= 0
(PDE6) 120597
120597119905(119890minus2119891 120597119871
1205971199093)+
120597119871
1205971199103= 0
(21)
such that the differential equations (21) are named conformalEuler-Lagrange equations on conformally flat manifold whichis shown in the form of (119879119872 119892 nabla 119869) Also therefore the triple(119879119872Φ
119871 119881) is called a conformal-Lagrangian mechanical
system on (119879119872 119892 nabla 119869)
7 Weyl-Euler-Lagrangian Equations forConservative Dynamical Systems
Proposition 26 We choose 119865 = i119881 119892 = Φ
119871 and 120582 = 2119891
at (11) and by considering (4) we can write Weyl-Lagrangiandynamic equation as follows
i119881(1198902119891
Φ119871) = i119881(Φ119871) minus 119889 (2119891) (22)
The second part (11) according to the law of conservation ofenergy [32] will not change for conservative dynamical systemsand i119881(Φ119871) = Φ
119871(119881)
Φ119871(119881) minus 2119889119891 = 119889119864
119871
Φ119871(119881) = 119889119864
119871+ 2119889119891 = 119889 (119864
119871+ 2119891)
(23)
From (21) above 119871 rarr 119871 + 2119891 So we can write
(PDE7) minus cos (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199101
)
minus sin (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199102
)
+120597 (119871 + 2119891)
1205971199091= 0
(PDE8) sin (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199101
)
minus cos (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199102
)
+120597 (119871 + 2119891)
1205971199092= 0
(PDE9) minus120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199103
)+120597 (119871 + 2119891)
1205971199093= 0
(PDE10) cos (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199091)
minus sin (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199092)
+120597 (119871 + 2119891)
1205971199101= 0
(PDE11) sin (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199091)
+ cos (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199092)
+120597 (119871 + 2119891)
1205971199102= 0
(PDE12) 120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199093)+
120597 (119871 + 2119891)1205971199103
= 0
(24)
and these differential equations (24) are named Weyl-Euler-Lagrange equations for conservative dynamical systems whichare constructed on conformally flat manifold (119879119872 119892 nabla 119869 119865)
and therefore the triple (119879119872Φ119871 119881) is called a Weyl-
Lagrangian mechanical system
8 Equations Solving with Computer
Theequations systems (21) and (24) have been solved by usingthe symbolic Algebra software and implicit solution is below
119871 (1199091 1199092 1199093 1199101 1199102 1199103 119905)
= exp (minus119894 lowast 119905) lowast 1198651 (1199103 minus 119894 lowast 1199093) + 1198652 (119905)
+ exp (119905 lowast 119894) lowast 1198653 (1199103 +1199093 lowast 119894) for 119891 = 0
(25)
10 Advances in Mathematical Physics
05
1
05 1 15 2minus1
minus1
minus05
minus05
(a)
05
1
05 1 15 2minus1
minus1
minus05
minus05
(b)
Figure 1
It is well known that an electromagnetic field is a physical fieldproduced by electrically charged objects The movement ofobjects in electrical magnetic and gravitational fields force isvery important For instance on a weather map the surfacewind velocity is defined by assigning a vector to each pointon a map So each vector represents the speed and directionof the movement of air at that point
The location of each object in space is represented bythree dimensions in physical space The dimensions whichare represented by higher dimensions are time positionmass and so forth The number of dimensions of (25) will bereduced to three and behind the graphics will be drawn Firstimplicit function at (25) will be selected as special After thefigure of (25) has been drawn for the route of the movementof objects in the electromagnetic field
Example 27 Consider
119871 (1199091 1199092 1199093 1199101 1199102 1199103 119905) = exp (minus119894 lowast 119905) + exp (119905 lowast 119894) lowast 119905 minus 1199052 (26)
(see Figure 1)
9 Discussion
A classical field theory explains the study of how one or morephysical fields interact with matter which is used in quantumand classical mechanics of physics branches In this study theEuler-Lagrange mechanical equations (21) and (24) derivedon a generalized on flat manifolds may be suggested to dealwith problems in electrical magnetic and gravitational fieldsforce for the path of movement (26) of defined space movingobjects [24]
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the agency BAP of PamukkaleUniversity
References
[1] W K Liu and S Jun Computational Nanomechanics of Mate-rials American Scientific Publishers Stevenson Ranch CalifUSA 2005
[2] M Tekkoyun and Y Yayli ldquoMechanical systems on generalized-quaternionic Kahler manifoldsrdquo International Journal of Geo-metric Methods in Modern Physics vol 8 no 7 pp 1419ndash14312011
[3] D B Fogel Epistemology of a theory of everything Weyl Ein-stein and the unification of physics [PhD thesis] GraduateSchool of theUniversity ofNotreDameNotreDame Ind USA2008
[4] Z Kasap and M Tekkoyun ldquoMechanical systems on almostparapseudo-KahlerndashWeyl manifoldsrdquo International Journal ofGeometric Methods in Modern Physics vol 10 no 5 Article ID1350008 8 pages 2013
[5] Z Kasap ldquoWeyl-mechanical systems on tangent manifoldsof constant 119882-sectional curvaturerdquo International Journal ofGeometric Methods in Modern Physics vol 10 no 10 Article ID1350053 pp 1ndash13 2013
[6] M Kapovich ldquoFlat conformal structures on 3-manifolds I uni-formization of closed seifert manifoldsrdquo Journal of DifferentialGeometry vol 38 no 1 pp 191ndash215 1993
[7] F Schwartz ldquoA volumetric Penrose inequality for conformallyflat manifoldsrdquo Annales Henri Poincare vol 12 no 1 pp 67ndash762011
[8] R S Kulkarni ldquoConformally flat manifoldsrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 69 pp 2675ndash2676 1972
[9] I G Dotti and R J Miatello ldquoOn the cohomology ring offlat manifolds with a special structurerdquo Revista De La Uni OnMatematica Argentina vol 46 no 2 pp 133ndash147 2005
Advances in Mathematical Physics 11
[10] A Szczepanski Kahler at Manifolds of Low Dimensions InstitutdesHautes Etudes Scientifiques Bures-sur-Yvette France 2005
[11] R Bartnik ldquoThemass of an asymptotically flat manifoldrdquo Com-munications on Pure and AppliedMathematics vol 39 no 5 pp661ndash693 1986
[12] M D M Gonzalez ldquoSingular sets of a class of locally confor-mally flat manifoldsrdquo Duke Mathematical Journal vol 129 no3 pp 551ndash572 2005
[13] S Akbulut and M Kalafat ldquoA class of locally conformally flat4-manifoldsrdquoNewYork Journal of Mathematics vol 18 pp 733ndash763 2012
[14] S-H Zhu ldquoThe classification of complete locally conformallyflat manifolds of nonnegative Ricci curvaturerdquo Pacific Journalof Mathematics vol 163 no 1 pp 189ndash199 1994
[15] H M Abood ldquoAlmost Hermitian manifold with flat Bochnertensorrdquo European Journal of Pure and Applied Mathematics vol3 no 4 pp 730ndash736 2010
[16] K Olszak and Z Olszak ldquoOn 4-dimensional conformally flatalmost 120576-Kahlerianmanifoldsrdquo Journal of Geometry and Physicsvol 62 no 5 pp 1108ndash1113 2012
[17] S Upadhyay ldquoA bounding question for almost flat manifoldsrdquoTransactions of the AmericanMathematical Society vol 353 no3 pp 963ndash972 2001
[18] 2015 httpmathworldwolframcomFlatManifoldhtml[19] 2015 httpenwikipediaorgwikiConformally flat manifold[20] D McDu and D Salamon J-Holomorphic Curves and Quantum
Cohomology AMS 1995[21] 2015 httpenwikipediaorgwikiConformal class[22] G B Folland ldquoWeyl manifoldsrdquo Journal of Differential Geome-
try vol 4 pp 145ndash153 1970[23] L Kadosh Topics in weyl geometry [PhD thesis] University of
California Berkeley Calif USA 1996[24] H Weyl Space-Time-Matter Dover Publications 1922 Trans-
lated from the 4th German edition by H Brose MethuenLondon UK Dover Publications New York NY USA 1952
[25] P Gilkey and S Nikcevic ldquoKahler and para-Kahler curvatureWeyl manifoldsrdquo httparxivorgabs10114844
[26] H Pedersen Y S Poon and A Swann ldquoThe Einstein-Weylequations in complex and quaternionic geometryrdquo DifferentialGeometry and Its Applications vol 3 no 4 pp 309ndash321 1993
[27] P Gilkey and S Nikcevic ldquoKahler-Weylmanifolds of dimension4rdquo httparxivorgabs11094532
[28] M Brozos-Vazquez P Gilkey and E Merino ldquoGeometricrealizations of Kaehler and of para-Kaehler curvature modelsrdquoInternational Journal of Geometric Methods in Modern Physicsvol 7 no 3 pp 505ndash515 2010
[29] 2015 httpenwikipediaorgwikiWeyl tensor[30] R Miron D Hrimiuc H Shimada and S V SabauThe Geom-
etry of Hamilton and Lagrange Spaces Kluwer Academic Pub-lishers 2002
[31] J Klein ldquoEspaces variationnels et mecaniquerdquo Annales delrsquoInstitut Fourier vol 12 pp 1ndash124 1962
[32] M de Leon and P R RodriguesMethods of Differential Geom-etry in Analytical Mechanics North-Holland Elsevier Amster-dam The Netherlands 1989
[33] R Abraham J E Marsden and T Ratiu Manifolds TensorAnalysis and Applications Springer New York NY USA 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Advances in Mathematical Physics
+ 1198902119891 cos (1199093)
1205972119871
120597119910312059711991021198891199103
+ 21198902119891 cos (1199093)120597119891
1205971199103
120597119871
12059711991021198891199103)+119883
3(119890
2119891 1205972119871
120597119910231198891199103
+ 21198902119891120597119891
1205971199103
120597119871
12059711991031198891199103)
+1198841(minus119890minus2119891 cos (1199093)
1205972119871
120597119910312059711990911198891199103
+ 2119890minus2119891 cos (1199093)120597119891
1205971199103
120597119871
12059711990911198891199103
+ 119890minus2119891 sin (1199093)
1205972119871
120597119910312059711990921198891199103
minus 2119890minus2119891 sin (1199093)120597119891
1205971199103
120597119871
12059711990921198891199103)
+1198842(minus119890minus2119891 sin (1199093)
1205972119871
120597119910312059711990911198891199103
+ 2119890minus2119891 sin (1199093)120597119891
1205971199103
120597119871
12059711990911198891199103
minus 119890minus2119891 cos (1199093)
1205972119871
120597119910312059711990921198891199103
+ 2119890minus2119891 cos (1199093)120597119891
1205971199103
120597119871
12059711990921198891199103)+119884
3(minus
1205972119871
120597119910312059711990931198891199103
+ 2119890minus2119891120597119891
1205971199103
120597
12059711990931198891199103)minus
120597119871
12059711991031198891199103
(18)
Using (11) we get first equations as follows
1198831[minus119890
2119891 cos (1199093)1205972119871
120597119909112059711991011198891199091
minus 11989021198912
120597119891
1205971199091cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119909112059711991021198891199091
minus 11989021198912
120597119891
1205971199091sin (1199093)
120597119871
12059711991021198891199091]
+1198832[minus119890
2119891 cos (1199093)1205972119871
120597119909212059711991011198891199091
minus 11989021198912
120597119891
1205971199092cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119909212059711991021198891199091
minus 11989021198912
120597119891
1205971199092sin (1199093)
120597119871
12059711991021198891199091]
+1198833[minus119890
2119891 cos (1199093)1205972119871
120597119909312059711991011198891199091
minus 11989021198912
120597119891
1205971199093cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119909312059711991021198891199091
minus 11989021198912
120597119891
1205971199093sin (1199093)
120597119871
12059711991021198891199091]
+1198841[minus119890
2119891 cos (1199093)1205972119871
120597119910211198891199091
minus 11989021198912
120597119891
1205971199101cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119910112059711991021198891199091
minus 11989021198912
120597119891
1205971199101sin (1199093)
120597119871
12059711991021198891199091]
+1198842[minus119890
2119891 cos (1199093)1205972119871
120597119910212059711991011198891199091
minus 11989021198912
120597119891
1205971199102cos (1199093)
120597119871
12059711991011198891199091 minus 119890
2119891 sin (1199093)1205972119871
120597119910221198891199091
minus 11989021198912
120597119891
1205971199102sin (1199093)
120597119871
12059711991021198891199091]
+1198843[minus119890
2119891 cos (1199093)1205972119871
120597119910312059711991011198891199091
minus 11989021198912
120597119891
1205971199103cos (1199093)
120597119871
12059711991011198891199091
minus 1198902119891 sin (1199093)
1205972119871
120597119910312059711991021198891199091
minus 11989021198912
120597119891
1205971199103sin (1199093)
120597119871
12059711991021198891199091] = minus
120597119871
12059711990911198891199091
(19)
From here
minus cos (1199093) 119881(1198902119891 120597119871
1205971199101)minus sin (1199093) 119881(119890
2119891 120597119871
1205971199102)+
120597119871
1205971199091
= 0
(20)
Advances in Mathematical Physics 9
If we think of the curve 120572 for all equations as an integralcurve of 119881 that is 119881(120572) = (120597120597119905)(120572) we find the followingequations
(PDE1) minus cos (1199093)120597
120597119905(119890
2119891 120597119871
1205971199101)
minus sin (1199093)120597
120597119905(119890
2119891 120597119871
1205971199102)+
120597119871
1205971199091= 0
(PDE2) sin (1199093)120597
120597119905(119890
2119891 120597119871
1205971199101)
minus cos (1199093)120597
120597119905(119890
2119891 120597119871
1205971199102)+
120597119871
1205971199092= 0
(PDE3) minus120597
120597119905(119890
2119891 120597119871
1205971199103)+
120597119871
1205971199093= 0
(PDE4) cos (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199091)
minus sin (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199092)+
120597119871
1205971199101= 0
(PDE5) sin (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199091)
+ cos (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199092)+
120597119871
1205971199102= 0
(PDE6) 120597
120597119905(119890minus2119891 120597119871
1205971199093)+
120597119871
1205971199103= 0
(21)
such that the differential equations (21) are named conformalEuler-Lagrange equations on conformally flat manifold whichis shown in the form of (119879119872 119892 nabla 119869) Also therefore the triple(119879119872Φ
119871 119881) is called a conformal-Lagrangian mechanical
system on (119879119872 119892 nabla 119869)
7 Weyl-Euler-Lagrangian Equations forConservative Dynamical Systems
Proposition 26 We choose 119865 = i119881 119892 = Φ
119871 and 120582 = 2119891
at (11) and by considering (4) we can write Weyl-Lagrangiandynamic equation as follows
i119881(1198902119891
Φ119871) = i119881(Φ119871) minus 119889 (2119891) (22)
The second part (11) according to the law of conservation ofenergy [32] will not change for conservative dynamical systemsand i119881(Φ119871) = Φ
119871(119881)
Φ119871(119881) minus 2119889119891 = 119889119864
119871
Φ119871(119881) = 119889119864
119871+ 2119889119891 = 119889 (119864
119871+ 2119891)
(23)
From (21) above 119871 rarr 119871 + 2119891 So we can write
(PDE7) minus cos (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199101
)
minus sin (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199102
)
+120597 (119871 + 2119891)
1205971199091= 0
(PDE8) sin (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199101
)
minus cos (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199102
)
+120597 (119871 + 2119891)
1205971199092= 0
(PDE9) minus120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199103
)+120597 (119871 + 2119891)
1205971199093= 0
(PDE10) cos (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199091)
minus sin (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199092)
+120597 (119871 + 2119891)
1205971199101= 0
(PDE11) sin (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199091)
+ cos (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199092)
+120597 (119871 + 2119891)
1205971199102= 0
(PDE12) 120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199093)+
120597 (119871 + 2119891)1205971199103
= 0
(24)
and these differential equations (24) are named Weyl-Euler-Lagrange equations for conservative dynamical systems whichare constructed on conformally flat manifold (119879119872 119892 nabla 119869 119865)
and therefore the triple (119879119872Φ119871 119881) is called a Weyl-
Lagrangian mechanical system
8 Equations Solving with Computer
Theequations systems (21) and (24) have been solved by usingthe symbolic Algebra software and implicit solution is below
119871 (1199091 1199092 1199093 1199101 1199102 1199103 119905)
= exp (minus119894 lowast 119905) lowast 1198651 (1199103 minus 119894 lowast 1199093) + 1198652 (119905)
+ exp (119905 lowast 119894) lowast 1198653 (1199103 +1199093 lowast 119894) for 119891 = 0
(25)
10 Advances in Mathematical Physics
05
1
05 1 15 2minus1
minus1
minus05
minus05
(a)
05
1
05 1 15 2minus1
minus1
minus05
minus05
(b)
Figure 1
It is well known that an electromagnetic field is a physical fieldproduced by electrically charged objects The movement ofobjects in electrical magnetic and gravitational fields force isvery important For instance on a weather map the surfacewind velocity is defined by assigning a vector to each pointon a map So each vector represents the speed and directionof the movement of air at that point
The location of each object in space is represented bythree dimensions in physical space The dimensions whichare represented by higher dimensions are time positionmass and so forth The number of dimensions of (25) will bereduced to three and behind the graphics will be drawn Firstimplicit function at (25) will be selected as special After thefigure of (25) has been drawn for the route of the movementof objects in the electromagnetic field
Example 27 Consider
119871 (1199091 1199092 1199093 1199101 1199102 1199103 119905) = exp (minus119894 lowast 119905) + exp (119905 lowast 119894) lowast 119905 minus 1199052 (26)
(see Figure 1)
9 Discussion
A classical field theory explains the study of how one or morephysical fields interact with matter which is used in quantumand classical mechanics of physics branches In this study theEuler-Lagrange mechanical equations (21) and (24) derivedon a generalized on flat manifolds may be suggested to dealwith problems in electrical magnetic and gravitational fieldsforce for the path of movement (26) of defined space movingobjects [24]
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the agency BAP of PamukkaleUniversity
References
[1] W K Liu and S Jun Computational Nanomechanics of Mate-rials American Scientific Publishers Stevenson Ranch CalifUSA 2005
[2] M Tekkoyun and Y Yayli ldquoMechanical systems on generalized-quaternionic Kahler manifoldsrdquo International Journal of Geo-metric Methods in Modern Physics vol 8 no 7 pp 1419ndash14312011
[3] D B Fogel Epistemology of a theory of everything Weyl Ein-stein and the unification of physics [PhD thesis] GraduateSchool of theUniversity ofNotreDameNotreDame Ind USA2008
[4] Z Kasap and M Tekkoyun ldquoMechanical systems on almostparapseudo-KahlerndashWeyl manifoldsrdquo International Journal ofGeometric Methods in Modern Physics vol 10 no 5 Article ID1350008 8 pages 2013
[5] Z Kasap ldquoWeyl-mechanical systems on tangent manifoldsof constant 119882-sectional curvaturerdquo International Journal ofGeometric Methods in Modern Physics vol 10 no 10 Article ID1350053 pp 1ndash13 2013
[6] M Kapovich ldquoFlat conformal structures on 3-manifolds I uni-formization of closed seifert manifoldsrdquo Journal of DifferentialGeometry vol 38 no 1 pp 191ndash215 1993
[7] F Schwartz ldquoA volumetric Penrose inequality for conformallyflat manifoldsrdquo Annales Henri Poincare vol 12 no 1 pp 67ndash762011
[8] R S Kulkarni ldquoConformally flat manifoldsrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 69 pp 2675ndash2676 1972
[9] I G Dotti and R J Miatello ldquoOn the cohomology ring offlat manifolds with a special structurerdquo Revista De La Uni OnMatematica Argentina vol 46 no 2 pp 133ndash147 2005
Advances in Mathematical Physics 11
[10] A Szczepanski Kahler at Manifolds of Low Dimensions InstitutdesHautes Etudes Scientifiques Bures-sur-Yvette France 2005
[11] R Bartnik ldquoThemass of an asymptotically flat manifoldrdquo Com-munications on Pure and AppliedMathematics vol 39 no 5 pp661ndash693 1986
[12] M D M Gonzalez ldquoSingular sets of a class of locally confor-mally flat manifoldsrdquo Duke Mathematical Journal vol 129 no3 pp 551ndash572 2005
[13] S Akbulut and M Kalafat ldquoA class of locally conformally flat4-manifoldsrdquoNewYork Journal of Mathematics vol 18 pp 733ndash763 2012
[14] S-H Zhu ldquoThe classification of complete locally conformallyflat manifolds of nonnegative Ricci curvaturerdquo Pacific Journalof Mathematics vol 163 no 1 pp 189ndash199 1994
[15] H M Abood ldquoAlmost Hermitian manifold with flat Bochnertensorrdquo European Journal of Pure and Applied Mathematics vol3 no 4 pp 730ndash736 2010
[16] K Olszak and Z Olszak ldquoOn 4-dimensional conformally flatalmost 120576-Kahlerianmanifoldsrdquo Journal of Geometry and Physicsvol 62 no 5 pp 1108ndash1113 2012
[17] S Upadhyay ldquoA bounding question for almost flat manifoldsrdquoTransactions of the AmericanMathematical Society vol 353 no3 pp 963ndash972 2001
[18] 2015 httpmathworldwolframcomFlatManifoldhtml[19] 2015 httpenwikipediaorgwikiConformally flat manifold[20] D McDu and D Salamon J-Holomorphic Curves and Quantum
Cohomology AMS 1995[21] 2015 httpenwikipediaorgwikiConformal class[22] G B Folland ldquoWeyl manifoldsrdquo Journal of Differential Geome-
try vol 4 pp 145ndash153 1970[23] L Kadosh Topics in weyl geometry [PhD thesis] University of
California Berkeley Calif USA 1996[24] H Weyl Space-Time-Matter Dover Publications 1922 Trans-
lated from the 4th German edition by H Brose MethuenLondon UK Dover Publications New York NY USA 1952
[25] P Gilkey and S Nikcevic ldquoKahler and para-Kahler curvatureWeyl manifoldsrdquo httparxivorgabs10114844
[26] H Pedersen Y S Poon and A Swann ldquoThe Einstein-Weylequations in complex and quaternionic geometryrdquo DifferentialGeometry and Its Applications vol 3 no 4 pp 309ndash321 1993
[27] P Gilkey and S Nikcevic ldquoKahler-Weylmanifolds of dimension4rdquo httparxivorgabs11094532
[28] M Brozos-Vazquez P Gilkey and E Merino ldquoGeometricrealizations of Kaehler and of para-Kaehler curvature modelsrdquoInternational Journal of Geometric Methods in Modern Physicsvol 7 no 3 pp 505ndash515 2010
[29] 2015 httpenwikipediaorgwikiWeyl tensor[30] R Miron D Hrimiuc H Shimada and S V SabauThe Geom-
etry of Hamilton and Lagrange Spaces Kluwer Academic Pub-lishers 2002
[31] J Klein ldquoEspaces variationnels et mecaniquerdquo Annales delrsquoInstitut Fourier vol 12 pp 1ndash124 1962
[32] M de Leon and P R RodriguesMethods of Differential Geom-etry in Analytical Mechanics North-Holland Elsevier Amster-dam The Netherlands 1989
[33] R Abraham J E Marsden and T Ratiu Manifolds TensorAnalysis and Applications Springer New York NY USA 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 9
If we think of the curve 120572 for all equations as an integralcurve of 119881 that is 119881(120572) = (120597120597119905)(120572) we find the followingequations
(PDE1) minus cos (1199093)120597
120597119905(119890
2119891 120597119871
1205971199101)
minus sin (1199093)120597
120597119905(119890
2119891 120597119871
1205971199102)+
120597119871
1205971199091= 0
(PDE2) sin (1199093)120597
120597119905(119890
2119891 120597119871
1205971199101)
minus cos (1199093)120597
120597119905(119890
2119891 120597119871
1205971199102)+
120597119871
1205971199092= 0
(PDE3) minus120597
120597119905(119890
2119891 120597119871
1205971199103)+
120597119871
1205971199093= 0
(PDE4) cos (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199091)
minus sin (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199092)+
120597119871
1205971199101= 0
(PDE5) sin (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199091)
+ cos (1199093)120597
120597119905(119890minus2119891 120597119871
1205971199092)+
120597119871
1205971199102= 0
(PDE6) 120597
120597119905(119890minus2119891 120597119871
1205971199093)+
120597119871
1205971199103= 0
(21)
such that the differential equations (21) are named conformalEuler-Lagrange equations on conformally flat manifold whichis shown in the form of (119879119872 119892 nabla 119869) Also therefore the triple(119879119872Φ
119871 119881) is called a conformal-Lagrangian mechanical
system on (119879119872 119892 nabla 119869)
7 Weyl-Euler-Lagrangian Equations forConservative Dynamical Systems
Proposition 26 We choose 119865 = i119881 119892 = Φ
119871 and 120582 = 2119891
at (11) and by considering (4) we can write Weyl-Lagrangiandynamic equation as follows
i119881(1198902119891
Φ119871) = i119881(Φ119871) minus 119889 (2119891) (22)
The second part (11) according to the law of conservation ofenergy [32] will not change for conservative dynamical systemsand i119881(Φ119871) = Φ
119871(119881)
Φ119871(119881) minus 2119889119891 = 119889119864
119871
Φ119871(119881) = 119889119864
119871+ 2119889119891 = 119889 (119864
119871+ 2119891)
(23)
From (21) above 119871 rarr 119871 + 2119891 So we can write
(PDE7) minus cos (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199101
)
minus sin (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199102
)
+120597 (119871 + 2119891)
1205971199091= 0
(PDE8) sin (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199101
)
minus cos (1199093)120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199102
)
+120597 (119871 + 2119891)
1205971199092= 0
(PDE9) minus120597
120597119905(119890
2119891 120597 (119871 + 2119891)1205971199103
)+120597 (119871 + 2119891)
1205971199093= 0
(PDE10) cos (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199091)
minus sin (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199092)
+120597 (119871 + 2119891)
1205971199101= 0
(PDE11) sin (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199091)
+ cos (1199093)120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199092)
+120597 (119871 + 2119891)
1205971199102= 0
(PDE12) 120597
120597119905(119890minus2119891 120597 (119871 + 2119891)
1205971199093)+
120597 (119871 + 2119891)1205971199103
= 0
(24)
and these differential equations (24) are named Weyl-Euler-Lagrange equations for conservative dynamical systems whichare constructed on conformally flat manifold (119879119872 119892 nabla 119869 119865)
and therefore the triple (119879119872Φ119871 119881) is called a Weyl-
Lagrangian mechanical system
8 Equations Solving with Computer
Theequations systems (21) and (24) have been solved by usingthe symbolic Algebra software and implicit solution is below
119871 (1199091 1199092 1199093 1199101 1199102 1199103 119905)
= exp (minus119894 lowast 119905) lowast 1198651 (1199103 minus 119894 lowast 1199093) + 1198652 (119905)
+ exp (119905 lowast 119894) lowast 1198653 (1199103 +1199093 lowast 119894) for 119891 = 0
(25)
10 Advances in Mathematical Physics
05
1
05 1 15 2minus1
minus1
minus05
minus05
(a)
05
1
05 1 15 2minus1
minus1
minus05
minus05
(b)
Figure 1
It is well known that an electromagnetic field is a physical fieldproduced by electrically charged objects The movement ofobjects in electrical magnetic and gravitational fields force isvery important For instance on a weather map the surfacewind velocity is defined by assigning a vector to each pointon a map So each vector represents the speed and directionof the movement of air at that point
The location of each object in space is represented bythree dimensions in physical space The dimensions whichare represented by higher dimensions are time positionmass and so forth The number of dimensions of (25) will bereduced to three and behind the graphics will be drawn Firstimplicit function at (25) will be selected as special After thefigure of (25) has been drawn for the route of the movementof objects in the electromagnetic field
Example 27 Consider
119871 (1199091 1199092 1199093 1199101 1199102 1199103 119905) = exp (minus119894 lowast 119905) + exp (119905 lowast 119894) lowast 119905 minus 1199052 (26)
(see Figure 1)
9 Discussion
A classical field theory explains the study of how one or morephysical fields interact with matter which is used in quantumand classical mechanics of physics branches In this study theEuler-Lagrange mechanical equations (21) and (24) derivedon a generalized on flat manifolds may be suggested to dealwith problems in electrical magnetic and gravitational fieldsforce for the path of movement (26) of defined space movingobjects [24]
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the agency BAP of PamukkaleUniversity
References
[1] W K Liu and S Jun Computational Nanomechanics of Mate-rials American Scientific Publishers Stevenson Ranch CalifUSA 2005
[2] M Tekkoyun and Y Yayli ldquoMechanical systems on generalized-quaternionic Kahler manifoldsrdquo International Journal of Geo-metric Methods in Modern Physics vol 8 no 7 pp 1419ndash14312011
[3] D B Fogel Epistemology of a theory of everything Weyl Ein-stein and the unification of physics [PhD thesis] GraduateSchool of theUniversity ofNotreDameNotreDame Ind USA2008
[4] Z Kasap and M Tekkoyun ldquoMechanical systems on almostparapseudo-KahlerndashWeyl manifoldsrdquo International Journal ofGeometric Methods in Modern Physics vol 10 no 5 Article ID1350008 8 pages 2013
[5] Z Kasap ldquoWeyl-mechanical systems on tangent manifoldsof constant 119882-sectional curvaturerdquo International Journal ofGeometric Methods in Modern Physics vol 10 no 10 Article ID1350053 pp 1ndash13 2013
[6] M Kapovich ldquoFlat conformal structures on 3-manifolds I uni-formization of closed seifert manifoldsrdquo Journal of DifferentialGeometry vol 38 no 1 pp 191ndash215 1993
[7] F Schwartz ldquoA volumetric Penrose inequality for conformallyflat manifoldsrdquo Annales Henri Poincare vol 12 no 1 pp 67ndash762011
[8] R S Kulkarni ldquoConformally flat manifoldsrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 69 pp 2675ndash2676 1972
[9] I G Dotti and R J Miatello ldquoOn the cohomology ring offlat manifolds with a special structurerdquo Revista De La Uni OnMatematica Argentina vol 46 no 2 pp 133ndash147 2005
Advances in Mathematical Physics 11
[10] A Szczepanski Kahler at Manifolds of Low Dimensions InstitutdesHautes Etudes Scientifiques Bures-sur-Yvette France 2005
[11] R Bartnik ldquoThemass of an asymptotically flat manifoldrdquo Com-munications on Pure and AppliedMathematics vol 39 no 5 pp661ndash693 1986
[12] M D M Gonzalez ldquoSingular sets of a class of locally confor-mally flat manifoldsrdquo Duke Mathematical Journal vol 129 no3 pp 551ndash572 2005
[13] S Akbulut and M Kalafat ldquoA class of locally conformally flat4-manifoldsrdquoNewYork Journal of Mathematics vol 18 pp 733ndash763 2012
[14] S-H Zhu ldquoThe classification of complete locally conformallyflat manifolds of nonnegative Ricci curvaturerdquo Pacific Journalof Mathematics vol 163 no 1 pp 189ndash199 1994
[15] H M Abood ldquoAlmost Hermitian manifold with flat Bochnertensorrdquo European Journal of Pure and Applied Mathematics vol3 no 4 pp 730ndash736 2010
[16] K Olszak and Z Olszak ldquoOn 4-dimensional conformally flatalmost 120576-Kahlerianmanifoldsrdquo Journal of Geometry and Physicsvol 62 no 5 pp 1108ndash1113 2012
[17] S Upadhyay ldquoA bounding question for almost flat manifoldsrdquoTransactions of the AmericanMathematical Society vol 353 no3 pp 963ndash972 2001
[18] 2015 httpmathworldwolframcomFlatManifoldhtml[19] 2015 httpenwikipediaorgwikiConformally flat manifold[20] D McDu and D Salamon J-Holomorphic Curves and Quantum
Cohomology AMS 1995[21] 2015 httpenwikipediaorgwikiConformal class[22] G B Folland ldquoWeyl manifoldsrdquo Journal of Differential Geome-
try vol 4 pp 145ndash153 1970[23] L Kadosh Topics in weyl geometry [PhD thesis] University of
California Berkeley Calif USA 1996[24] H Weyl Space-Time-Matter Dover Publications 1922 Trans-
lated from the 4th German edition by H Brose MethuenLondon UK Dover Publications New York NY USA 1952
[25] P Gilkey and S Nikcevic ldquoKahler and para-Kahler curvatureWeyl manifoldsrdquo httparxivorgabs10114844
[26] H Pedersen Y S Poon and A Swann ldquoThe Einstein-Weylequations in complex and quaternionic geometryrdquo DifferentialGeometry and Its Applications vol 3 no 4 pp 309ndash321 1993
[27] P Gilkey and S Nikcevic ldquoKahler-Weylmanifolds of dimension4rdquo httparxivorgabs11094532
[28] M Brozos-Vazquez P Gilkey and E Merino ldquoGeometricrealizations of Kaehler and of para-Kaehler curvature modelsrdquoInternational Journal of Geometric Methods in Modern Physicsvol 7 no 3 pp 505ndash515 2010
[29] 2015 httpenwikipediaorgwikiWeyl tensor[30] R Miron D Hrimiuc H Shimada and S V SabauThe Geom-
etry of Hamilton and Lagrange Spaces Kluwer Academic Pub-lishers 2002
[31] J Klein ldquoEspaces variationnels et mecaniquerdquo Annales delrsquoInstitut Fourier vol 12 pp 1ndash124 1962
[32] M de Leon and P R RodriguesMethods of Differential Geom-etry in Analytical Mechanics North-Holland Elsevier Amster-dam The Netherlands 1989
[33] R Abraham J E Marsden and T Ratiu Manifolds TensorAnalysis and Applications Springer New York NY USA 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Advances in Mathematical Physics
05
1
05 1 15 2minus1
minus1
minus05
minus05
(a)
05
1
05 1 15 2minus1
minus1
minus05
minus05
(b)
Figure 1
It is well known that an electromagnetic field is a physical fieldproduced by electrically charged objects The movement ofobjects in electrical magnetic and gravitational fields force isvery important For instance on a weather map the surfacewind velocity is defined by assigning a vector to each pointon a map So each vector represents the speed and directionof the movement of air at that point
The location of each object in space is represented bythree dimensions in physical space The dimensions whichare represented by higher dimensions are time positionmass and so forth The number of dimensions of (25) will bereduced to three and behind the graphics will be drawn Firstimplicit function at (25) will be selected as special After thefigure of (25) has been drawn for the route of the movementof objects in the electromagnetic field
Example 27 Consider
119871 (1199091 1199092 1199093 1199101 1199102 1199103 119905) = exp (minus119894 lowast 119905) + exp (119905 lowast 119894) lowast 119905 minus 1199052 (26)
(see Figure 1)
9 Discussion
A classical field theory explains the study of how one or morephysical fields interact with matter which is used in quantumand classical mechanics of physics branches In this study theEuler-Lagrange mechanical equations (21) and (24) derivedon a generalized on flat manifolds may be suggested to dealwith problems in electrical magnetic and gravitational fieldsforce for the path of movement (26) of defined space movingobjects [24]
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the agency BAP of PamukkaleUniversity
References
[1] W K Liu and S Jun Computational Nanomechanics of Mate-rials American Scientific Publishers Stevenson Ranch CalifUSA 2005
[2] M Tekkoyun and Y Yayli ldquoMechanical systems on generalized-quaternionic Kahler manifoldsrdquo International Journal of Geo-metric Methods in Modern Physics vol 8 no 7 pp 1419ndash14312011
[3] D B Fogel Epistemology of a theory of everything Weyl Ein-stein and the unification of physics [PhD thesis] GraduateSchool of theUniversity ofNotreDameNotreDame Ind USA2008
[4] Z Kasap and M Tekkoyun ldquoMechanical systems on almostparapseudo-KahlerndashWeyl manifoldsrdquo International Journal ofGeometric Methods in Modern Physics vol 10 no 5 Article ID1350008 8 pages 2013
[5] Z Kasap ldquoWeyl-mechanical systems on tangent manifoldsof constant 119882-sectional curvaturerdquo International Journal ofGeometric Methods in Modern Physics vol 10 no 10 Article ID1350053 pp 1ndash13 2013
[6] M Kapovich ldquoFlat conformal structures on 3-manifolds I uni-formization of closed seifert manifoldsrdquo Journal of DifferentialGeometry vol 38 no 1 pp 191ndash215 1993
[7] F Schwartz ldquoA volumetric Penrose inequality for conformallyflat manifoldsrdquo Annales Henri Poincare vol 12 no 1 pp 67ndash762011
[8] R S Kulkarni ldquoConformally flat manifoldsrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 69 pp 2675ndash2676 1972
[9] I G Dotti and R J Miatello ldquoOn the cohomology ring offlat manifolds with a special structurerdquo Revista De La Uni OnMatematica Argentina vol 46 no 2 pp 133ndash147 2005
Advances in Mathematical Physics 11
[10] A Szczepanski Kahler at Manifolds of Low Dimensions InstitutdesHautes Etudes Scientifiques Bures-sur-Yvette France 2005
[11] R Bartnik ldquoThemass of an asymptotically flat manifoldrdquo Com-munications on Pure and AppliedMathematics vol 39 no 5 pp661ndash693 1986
[12] M D M Gonzalez ldquoSingular sets of a class of locally confor-mally flat manifoldsrdquo Duke Mathematical Journal vol 129 no3 pp 551ndash572 2005
[13] S Akbulut and M Kalafat ldquoA class of locally conformally flat4-manifoldsrdquoNewYork Journal of Mathematics vol 18 pp 733ndash763 2012
[14] S-H Zhu ldquoThe classification of complete locally conformallyflat manifolds of nonnegative Ricci curvaturerdquo Pacific Journalof Mathematics vol 163 no 1 pp 189ndash199 1994
[15] H M Abood ldquoAlmost Hermitian manifold with flat Bochnertensorrdquo European Journal of Pure and Applied Mathematics vol3 no 4 pp 730ndash736 2010
[16] K Olszak and Z Olszak ldquoOn 4-dimensional conformally flatalmost 120576-Kahlerianmanifoldsrdquo Journal of Geometry and Physicsvol 62 no 5 pp 1108ndash1113 2012
[17] S Upadhyay ldquoA bounding question for almost flat manifoldsrdquoTransactions of the AmericanMathematical Society vol 353 no3 pp 963ndash972 2001
[18] 2015 httpmathworldwolframcomFlatManifoldhtml[19] 2015 httpenwikipediaorgwikiConformally flat manifold[20] D McDu and D Salamon J-Holomorphic Curves and Quantum
Cohomology AMS 1995[21] 2015 httpenwikipediaorgwikiConformal class[22] G B Folland ldquoWeyl manifoldsrdquo Journal of Differential Geome-
try vol 4 pp 145ndash153 1970[23] L Kadosh Topics in weyl geometry [PhD thesis] University of
California Berkeley Calif USA 1996[24] H Weyl Space-Time-Matter Dover Publications 1922 Trans-
lated from the 4th German edition by H Brose MethuenLondon UK Dover Publications New York NY USA 1952
[25] P Gilkey and S Nikcevic ldquoKahler and para-Kahler curvatureWeyl manifoldsrdquo httparxivorgabs10114844
[26] H Pedersen Y S Poon and A Swann ldquoThe Einstein-Weylequations in complex and quaternionic geometryrdquo DifferentialGeometry and Its Applications vol 3 no 4 pp 309ndash321 1993
[27] P Gilkey and S Nikcevic ldquoKahler-Weylmanifolds of dimension4rdquo httparxivorgabs11094532
[28] M Brozos-Vazquez P Gilkey and E Merino ldquoGeometricrealizations of Kaehler and of para-Kaehler curvature modelsrdquoInternational Journal of Geometric Methods in Modern Physicsvol 7 no 3 pp 505ndash515 2010
[29] 2015 httpenwikipediaorgwikiWeyl tensor[30] R Miron D Hrimiuc H Shimada and S V SabauThe Geom-
etry of Hamilton and Lagrange Spaces Kluwer Academic Pub-lishers 2002
[31] J Klein ldquoEspaces variationnels et mecaniquerdquo Annales delrsquoInstitut Fourier vol 12 pp 1ndash124 1962
[32] M de Leon and P R RodriguesMethods of Differential Geom-etry in Analytical Mechanics North-Holland Elsevier Amster-dam The Netherlands 1989
[33] R Abraham J E Marsden and T Ratiu Manifolds TensorAnalysis and Applications Springer New York NY USA 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 11
[10] A Szczepanski Kahler at Manifolds of Low Dimensions InstitutdesHautes Etudes Scientifiques Bures-sur-Yvette France 2005
[11] R Bartnik ldquoThemass of an asymptotically flat manifoldrdquo Com-munications on Pure and AppliedMathematics vol 39 no 5 pp661ndash693 1986
[12] M D M Gonzalez ldquoSingular sets of a class of locally confor-mally flat manifoldsrdquo Duke Mathematical Journal vol 129 no3 pp 551ndash572 2005
[13] S Akbulut and M Kalafat ldquoA class of locally conformally flat4-manifoldsrdquoNewYork Journal of Mathematics vol 18 pp 733ndash763 2012
[14] S-H Zhu ldquoThe classification of complete locally conformallyflat manifolds of nonnegative Ricci curvaturerdquo Pacific Journalof Mathematics vol 163 no 1 pp 189ndash199 1994
[15] H M Abood ldquoAlmost Hermitian manifold with flat Bochnertensorrdquo European Journal of Pure and Applied Mathematics vol3 no 4 pp 730ndash736 2010
[16] K Olszak and Z Olszak ldquoOn 4-dimensional conformally flatalmost 120576-Kahlerianmanifoldsrdquo Journal of Geometry and Physicsvol 62 no 5 pp 1108ndash1113 2012
[17] S Upadhyay ldquoA bounding question for almost flat manifoldsrdquoTransactions of the AmericanMathematical Society vol 353 no3 pp 963ndash972 2001
[18] 2015 httpmathworldwolframcomFlatManifoldhtml[19] 2015 httpenwikipediaorgwikiConformally flat manifold[20] D McDu and D Salamon J-Holomorphic Curves and Quantum
Cohomology AMS 1995[21] 2015 httpenwikipediaorgwikiConformal class[22] G B Folland ldquoWeyl manifoldsrdquo Journal of Differential Geome-
try vol 4 pp 145ndash153 1970[23] L Kadosh Topics in weyl geometry [PhD thesis] University of
California Berkeley Calif USA 1996[24] H Weyl Space-Time-Matter Dover Publications 1922 Trans-
lated from the 4th German edition by H Brose MethuenLondon UK Dover Publications New York NY USA 1952
[25] P Gilkey and S Nikcevic ldquoKahler and para-Kahler curvatureWeyl manifoldsrdquo httparxivorgabs10114844
[26] H Pedersen Y S Poon and A Swann ldquoThe Einstein-Weylequations in complex and quaternionic geometryrdquo DifferentialGeometry and Its Applications vol 3 no 4 pp 309ndash321 1993
[27] P Gilkey and S Nikcevic ldquoKahler-Weylmanifolds of dimension4rdquo httparxivorgabs11094532
[28] M Brozos-Vazquez P Gilkey and E Merino ldquoGeometricrealizations of Kaehler and of para-Kaehler curvature modelsrdquoInternational Journal of Geometric Methods in Modern Physicsvol 7 no 3 pp 505ndash515 2010
[29] 2015 httpenwikipediaorgwikiWeyl tensor[30] R Miron D Hrimiuc H Shimada and S V SabauThe Geom-
etry of Hamilton and Lagrange Spaces Kluwer Academic Pub-lishers 2002
[31] J Klein ldquoEspaces variationnels et mecaniquerdquo Annales delrsquoInstitut Fourier vol 12 pp 1ndash124 1962
[32] M de Leon and P R RodriguesMethods of Differential Geom-etry in Analytical Mechanics North-Holland Elsevier Amster-dam The Netherlands 1989
[33] R Abraham J E Marsden and T Ratiu Manifolds TensorAnalysis and Applications Springer New York NY USA 2001
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Differential EquationsInternational Journal of
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Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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