Differentially Constrained Dynamics

Wayne LawtonDepartment of Mathematics

National University of Singapore matwml@nus.edu.sg

(65)96314907

with monetary applications to rolling coins*

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*the methods described herein are intended for quantum gravitational speculation, the

author disclaims any financial responsibility for fools’ attempts to apply these methods in

Sentosa Casinos

Introduction

Our objective is to explain the physics behind some of the results in the paper Nonholonomic Dynamics by Anthony M. Bloch, Jerrold E. Marsden and Dmitry V. Zenkov,, Notices AMS, 52(3), 2005.

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Recall from the vufoils titled Connections.ppt that a distribution (in the sense of Frobenius) on an m-dim connected manifold M is defined by a smoothly varying subspace c(x) of the tangent space T_x(M) at x to M at every point x in M. We note that dim(c(x)) is constant and define p = m – dim(c).

Definition the Grassman manifold G(m,k) consists of all k-dim subpaces of R^m, it is the homogeneous spaceO(m)/(O(p)xO(m-p)) so has dimension p(m-p).

Remark c can be described a section of the bundle over M whose fibers is homeomorphic to G(m,p)

Distribution Formsc can be defined by a collection of p linearly independent.

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Tp ],...,[ 1

pivMTvxc ix ,...,1,0)(:)()(

then there exists a p x m matrix (valued function of x) E

.

If we introduce local coordinates.that has rank p and .

Tmxxx ],...,[ 1

dxE 1

an invertible p x p matrix and c is defined by the forms.

the coordinate indices so that.

][ CBEhence we may re-label

where B is.

dxABT

p

1

],...,1[ where.

][1CBIA

hence.

m

pj jijii pidxAdx1

,...,1,

differential 1-forms

.

Unconstrained Dynamics

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The dynamics of a system with kinetic energy T and forces F (with no constraints) is

.F

xT

where

.xxdt

dx

For conservative. x

VF

0xL

we have.

where we define the Lagrangian .

.VTL For local coordinates.

Tmxxx ],...,[ 1

we obtain m-equations and m-variables..

Holonomic Constraints

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such that

. .FxL

constant,...,constant1 pCCis to assume that the constraints are imposed by a constraint force F that is a differential 1-form that kills every vector that is tangent to the (m-p) dimensional submanifold of the tangent space of M at each point. This is equivalent to D’Alembert’s principle (forces of constraint can do no work to ‘virtual displacements’) and is equivalent to the existence of p variables .

p ,...,1

One method to develop the dynamics of a system with Lagrangian L that is subject to holonomic constraints

.

xCF ip

i i

1

The (m+p) variables (x’s & lambda’s) are computed from p constraint equations and the m-equations given by

.

Example: Particle on Inclined Plane

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where

.

tan1

1

111

1

xCxm

xL

222

2122121 )(),,,( mgxxxxxxxL m

and .

Here m = 2, p = 1 and for suitable coordinates

.tan),( 12211 xxxxC

is the (fixed) angle of the inclined plane. Therefore

.

12

112

2

xCmgxm

xL

0tan12 xx

and

. and

. 21 cosmg and

. sincos1 gx and

.

22 singx

Nonholonomic Constraints

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such that

.

.FxL

p ,...,1

For nonholonomic constraints D’Alemberts principle can also be applied to obtain the existence of

. ip

i iF

1

where the mu-forms describe the velocity constraints

. .,...,1,0)( pixi

The (m+p) variables (x’s & lambda’s) are computed from the p constraint equations above and the m-equations

Equivalent Form for Constraints

8

ppFxL

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Since the mu’s and omega’s give the same connection we can obtain an equivalent system of equations with different lambda’s (Lagrange multipliers)

.,...,1,0)( pixi On the following page we will show how to eliminate the Lagrange multipliers so as to reduce these equations to the form given in Equation (3) on page 326 of the Nonholomorphic Dynamics paper mentioned on page 2 of this paper.

Eliminating Lagrange Multipliers

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p

i

m

pjjijii

m

kk

k

dxAdxdxxL

1 11

We observe that we can express .

pii

i xL

,...,1, and reduced (m-p) equations

.

hence we solve for the Lagrange multipliers to obtain

xL

mpjxLA

xL p

ij

ikj

,...,1,1

which with

.the p constraint equations determine the m variables.

.

Let the Coins RollWe consider a more general form of the rolling coin problem described on p 62-64 of Analytical Mechanics by Louis N. Hand and Janet D. Finch. Here theta is the angle the radius R, mass m coin makes with the y-axis, phi is the rotation angle as it rolls along a surface described by the graph of the height function z(x,y). So for this problem m = 4 (variables phi, theta, x, y), p=2. ),(22

8122

43 yxmgzmRmRL

dRdx sin1

and constraints

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dRdy cos2

02241

mRL

yL

xLL RRmR

cossin22

23

),cos(sin yz

xzmgR

cos,sin RyRx Exercise compare with Hand-Finch solution on p 64

Ehresmann ConnectionLocally on M the 1-forms.

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pivMTvxc ix ,...,1,0)(:)()( define the distribution.

Hence they also define a fiber bundle .where.

m

pj jijii pidxAdx1

,...,1,

BE

pmm RR ,is an open subset of . ),...,(),...,,,...,( 111 ppppp xxxxxx

BE, and.

Therefore. )()(~ ETxc

)(xc can be identified with a horizontal.subspace

.and this describes.an Ehresmann connection

.BE c~ on

.

When are the constraints holonomic ?Our objective is to explain the relationship of this Ehresmann connection to the differentially constrained dynamics, in particular to prove the assertion, made on the top right of page 326 in paper by Bloch, Marsden and Zenkov, that its curvature tensor vanishes if and only if the distribution c is involutive or integrable. This means that the differential constraints are equivalent to holonomic constraints. Theorem 2.4 on page 82 of Lectures on Differential Geometry by S. S. Chern, W. H. Chen and K. S. Lam, this is equivalent to the condition

12

pid pii ,...,1),,...,( mod 0

Calculation

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where.

mkjp

pkjijk dxdxR

11 ),...,( mod

jxAp

kxA

xA

xAi

jk AAR ikij

j

ik

k

ij

1

m

pkkx

Ap

kkkx

Aj dxdx

k

ij

k

ij

11

),...,( mod 11

p

m

pjjji dxd

where.

pip ,...,1),,...,( mod 0 1 if and only if

0ijk011 id if and only if

Curved CoinsLet ‘s compute the curvature for the rolling coin system 4321 ,,, xxyxxx

4311 sinsin dxxRdxdRdx

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3243142313 cos,sin,0 xRAxRAAA

3134 cos xRR

4322 coscos dxxRdxdRdy

3234 sin xRR

Curvature

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Reference: Geometric Mechanics,Lagrangian Reduction and Nonholonomic Systems by H. Cendra, J. E. Marsden, and T. S. Ratiu, p. 221-273 in Mathematics Unlimited - 2001 and Beyond, Springer, 2001.

mkjpxkj

ijk i

dxdxR1

Exercise Show that the expression aboveequals the curvature, as defined on page 8 of Connections.ppt, of the Ehresmann connection.

c~

We observe that each fiber ofpR

has values in the tangent spaces to the fibers.

BE

is homeomorphic to

and that the 2-form on E