Geometric integration and static equilibrium states of

finite, nearest-neighbour-coupled lattice models.

S.C.Green†‡ C.J.Budd†‡¶ G.W.Hunt†§

October 28, 2010

Abstract

This paper presents new mathematical results which show how the static equilibriumstates of a general nearest-neighbour coupled lattice may be found using a discreteboundary value problem. These results use ideas from discrete mechanics to allow thedetermination of the detailed symmetry properties of the iterated map that defines thediscrete boundary value problem and also to model all types of boundary behaviourfor the finite lattice model. These mathematical results are then applied to two latticesystems. The first is a pin-jointed mechancial lattice system, for which we determinethe full bifurcation diagram of static equilibrium states for the lattice with six andsixteen links. The second is the Fermi-Pasta-Ulam lattice. It is shown that there is anexplosion of 3N static equilibrium states in the FPU lattice for 4β < α2 where α andβ are parameters of the FPU lattice and N + 1 is the number of lattice sites.

1 Introduction

One dimensional lattice models are of interest in a large number of scientific fields, for ex-ample study of DNA [], atomic lattices [], structural mechanics [7, 1, 8]. For some of theseapplications it is the time evolution of these lattices which is of interest, with substantial re-search effort concentrating on behaviour such as discrete breathers [3] and pattern formation[]. Some authors, however, concentrate on the static behaviour of these lattices [1, 8, 7].

Study of the static behaviour of a particular lattice model is equivalent to studying thestationary points of the potential energy of the lattice. The potential energy for a generalfinite, one dimensional, nearest neighbour coupled lattice can be written as

V (Q0, . . . , QN ) = h

N∑n=0

v(Qn) + h

N−1∑n=0

w

(Qn+1 −Qn

h

)+ a(Q0)− b(QN ) (1)

where Qn is the displacement coordinate for lattice site n, h is a parameter of the system,N + 1 is the number of lattice sites and the functions v and w are real functions of areal variable. If we interpret (1) in terms of the standard mass-spring model of an atomicchain shown in Figure 1, the derivative of the function v gives the restoring force eachmass feels towards its equilibrium position in the absence of any coupling springs, whilethe derivative of the function w gives the force displacement behaviour of the springs thatcouple neighbouring masses. For these reasons we call the function v the on-site potentialand the function w the coupling potential.

The equilibrium equations for this lattice are then given by ∂V/∂Qn = 0 for n = 0, . . . , N .It is known, for certain specific lattice models (for example [1, 8, 7]), that these equilibrliumequations can be written in the form of a two dimensional iterated map. The iterates of thismap are then used to find the static equilibrium configurations of the lattice. So far in the

†Centre for Nonlinear Mechanics, University of Bath, Bath, BA2 7AY, UK‡Department of Mathematical Sciences§Department of Mechanical Engineering¶Corresponding author; mascjb@bath.ac.uk

1

Figure 1: Mass spring model which often leads to a lattice model of the form (1) and is themotivation for the naming of the terms in equation (1).

literature these iterated maps have been derived on a case-by-case basis. It has been noticed,[1], that there is a certain amount of choice in the derivation of the discrete boundary valueproblem, and also claimed that modelling boundary conditions other than fixed lattice endpoints(Q0, QN fixed) is not possible.

In this paper, theory from the geometric numerical integration literature, specificiallydiscrete mechanics (see, for example [9]), is used to precisely derive an iterated map from therequirement that a potential energy function be stationary. This then leads to a result thatenables the modelling of all four (fixed-fixed, free-fixed, fixed-free and free-free) combinationsof end lattice behaviour using a discrete boundary value problem. This approach also shows,in detail, how the symmetries of the iterated maps are related to the different boundaryconditions that may be modelled. This link with geometric integration then also allows usto easily see the reasons behind the symmetries of the iterated map and to also derive acontinuum limit for each lattice model.

The mathematical results described above are presented in detail in Section 2. In sections3 and 4 these results are used to find new results on two lattice models from the literature.Section 3 considers the linked mechanical structure of [7] and extends their work to considerthe whole bifurcation diagram of the lattice’s static equlibrium states and to prove someof its properties. Section 4 presents some results on the famous Fermi-Pasta-Ulam (FPU)lattice model. Specifically it is shown that at 4β = α2 there is an explosion of 3N staticequilibrium states. In conclusion we summarise the theory and results obtained in thispaper.

2 Mathematical Results

2.1 Discrete mechanics

The theory of discrete mechanics (see for example [9]) is concerned with finding stationarypoints of functions of the form

Sd(Q) =N∑n=0

Ld(Qn, Qn+1) (2)

where Q ∈ RN+1 and Ld : R2 ⊇ ΩdL→ ΩrL

⊆ R. This theory can be seen to proceed byanalogy with classical mechanics which searches for stationary points of functionals of theform

S(q(t)) =∫ tf

t0

L(q(t), q(t)) dt (3)

where q : R→ R and L : R→ R. In classical mechanics equation (3) is know as the definitionof the action functional S and the function L is known as the Lagrangian. Similarly, in thetheory of discrete mechanics equation (2) is know as the definition of the discrete action andthe function Ld is the discrete Lagrangian.

In classical mechanics, requiring the action functional to be stationary leads to the Euler-Lagrange equation; a second order differential equation for actions of the form (3). Under theassumption of a regular Lagrangian (see Section 2.4) the Lagrangian function can be trans-formed into a Hamiltonian function via a Legendre transform. This Hamiltonian functionthen leads to a set of coupled first order differential equations, Hamilton’s equations.

2

Similarly, in discrete mechanics, requiring the action function to be stationary leads to thediscrete Euler-Lagrange equations; a second order difference equation for action functionsof the form (2). Under the assumption of a regular discrete Lagrangian (see [9, p23])the discrete Euler-Lagrange equation can be transformed into a set of coupled first orderdifference equations.

In this paper the problem of locating stationary points of the action function (2) ismodified slightly to give four related problems labelled ©1 , ©2 , ©3 and ©4 . In the followingwe use an abbreviated notation for the evaluation of the derivative of the function Ld,

D1Ld(Qn, Qn+1) =∂Ld(a, b)

∂a

∣∣∣∣a=Qnb=Qn+1

similarly, D2Ld(Qn, Qn+1) =∂Ld(a, b)

∂b

∣∣∣∣a=Qnb=Qn+1

.

Problem Set 2.1. Let a, b ∈ C1(R), Ld : ΩdL→ ΩrL

, ΩdL,ΩrL

⊆ R be at least oncedifferentiable with respect to each of its arguments, and :

1. for fixed Qn, D1Ld(Qn, Qn+1) : ΩdL→ ΩrL

is a surjection;

2. for fixed Qn+1, D2Ld(Qn, Qn+1) : ΩdL→ ΩrL

is a surjection;

3. R(ΩdL) = ΩdL

where R : R2 → R2, (x, y) 7→ (y, x);

4. N(ΩrL) = ΩrL

where N : R→ R, x 7→ −x.

Also define

ΩQ = Q = (Q0, . . . , QN ) ∈ RN+1 : (Qn, Qn+1) ∈ ΩdLfor all n = 0, . . . , N.

The fixed constants α0 and αN satisfy α0 ∈ P1(ΩdL) and α0 ∈ P2(ΩdL

) where P1, P2 : R2 →R such that P1 : (x, y) 7→ (x) and P1 : (x, y) 7→ (y). Then, subject to these conditions, findall of the stationary points, Q = (Q0, . . . , QN ) of the function Sd : ΩQ → R

Sd(Q) =N∑n=0

Ld(Qn, Qn+1) + a(Q0)− b(QN ) (4)

subject to one of the following sets of constraints

©1 Q0 = α0, QN = αN©2 Q0 = α0

©3 QN = αN©4 None.

In the above problem set the problem of finding stationary points of the function (2) hasbeen modified to allow for more diverse behaviour at the boundaries. Also, the restrictionthat the discrete Lagrangian be regular has been relaxed slightly. These modifications havebeen made so that these ideas can be used to find the static equilibrium states of nearestneighbour coupled lattices of the form (1) described in the introduction. Modifying thediscrete action also leads to modified coupled first order difference equations. The maintheorem of this section, Theorem 2.1 given below, proves that a set of variables Q0, . . . , QNis a solution to one of the problems in Problem Set 2.2 if and only if it is also a solution tothe corresponding problem in Problem Set 2.1 above. This is the main result that will beused in the next section to find the static equilibrium states of nearest neighbour coupledlattices.

Problem Set 2.2. Let Ld be the function Ld from Problem Set 2.1. Find all the vectorsQ = (Q0, . . . , QN ) and P = (P0, . . . , PN ) that satisfy the following discrete boundary valueproblem.

Xn =(QnPn

), Xn+1 = φ(Xn) n = 0, . . . , N − 1,

3

where φ : R× ΩrL→ R× ΩrL

, (Qn, Pn)T → (Qn+1, Pn+1)T is defined implicitly by

Pn = −D1Ld(Qn, Qn+1) (5)Pn+1 = D2Ld(Qn, Qn+1), (6)

and the the values of (Q0, P0) and (QN , PN ) are subject to one of the following sets of bound-ary conditions.

Boundary behaviour Q0 P0 QN PN

©1 α0 free αN free©2 α0 free free b′(QN )©3 free a′(Q0) αN free©4 free a′(Q0) free b′(QN ).

Theorem 2.1. The discrete boundary value problem described in Problem Set 2.2 is welldefined and a set of iterates, (Q0, Q1, . . . , QN ), is a solution to this discrete boundary valueproblem if and only if it also a solution to the corresponding problem in Problem Set 2.1.

Proof. Consider the first variation of the function S with respect to the variables Qn

δ1S =N−1∑i=0

D1Ld(Qi, Qi+1)δQi +N−1∑i=0

D2Ld(Qi, Qi+1)δQi+1

+ a′(Q0)δQ0 + b′(QN )δQN

=N−1∑i=1

(D1Ld(Qi, Qi+1) + D2Ld(Qi−1, Qi)

)δQi

+(D1Ld(Q0, Q1) + a′(Q0)

)δQ0 +

(D2Ld(QN−1, QN )− b′(QN )

)δQN .

In Problem Sets 2.1 and 2.2 constraints are only permitted on Q0 and QN and so δ1S = 0(and ∂S/∂Qn = 0 for all n) if and only if

D1Ld(Qi, Qi+1) + D2Ld(Qi−1, Qi) = 0 for i = 1, . . . , N − 1 (7)and D1Ld(Q0, Q1) + a′(Q0) = 0 or δQ0 = 0 (8)and D2Ld(QN−1, QN )− b′(QN ) = 0 or δQN = 0. (9)

Equations (7) are called the discrete Euler-Lagrange equations and, in this case, are a setof second order difference equations. Equations (8) and (9) are the boundary conditions forthis set of difference equations. We now prove that the second order difference equations(7) with the boundary conditions (8) and (9) are equivalent to the discrete boundary valueproblems given in Problem Set 2.2.

If we define Pi ∈ R by

Pi ≡ −D1Ld(Qi, Qi+1) for i = 0, . . . , N − 1

D2Ld(QN−1, QN ) for i = N(10)

then the discrete Euler-Lagrange equations (7) and (10) for i = N give

Pi+1 = D2Ld(Qi, Qi+1) for i = 0, . . . , N − 1. (11)

In terms of the new Pi variables, boundary condition (8) becomes P0 = a′(Q0) or Q0 = α0

and (9) becomes PN = b′(QN ) or QN = αN for some fixed constants α0 and αN .Conditions 1 and 4 from Problem Set 2.1 ensure that it is always possible to find at

least one Qi+1 that satisfies (10), given Qi and Pi, for i = 0, . . . , N − 1. Therefore (10) and(11) implicitly define (but not necessarily uniquely) a map φ : (Qi, Pi)T → (Qi+1, Pi+1)T

for i = 1, . . . , N − 1.Condition 2 from Problem Set 2.1 ensures that it is always possible to find at least

one Qi that satisfies (11), given Qi+1 and Pi+1, for i = 0, . . . , N − 1. Therefore (10) and

4

(11) implicitly define (but not necessarily uniquely) a backwards map φ : (Qi+1, Pi+1)T →(Qi, Pi)T for i = 1, . . . , N − 1.

Condition 3 from Problem Set 2.1 ensures that for the forwards (backward) map theQi+1 (Qi) found is within the range of the first (second) argument of Ld implying thatthe map from (Qi+1, Pi+1) → (Qi+2, Pi+2) ((Qi, Pi) → (Qi−1, Pi−1)) is also well defined.Letting Xi = (Qi, Pi)T for i = 0, . . . , N finishes the proof.

2.2 Application to nearest neighbour coupled lattices

As described in the introduction, the potential energy for a general lattice with N+1 latticesites and nearest neighbour coupling can be written as

V (Q0, . . . , QN ) = h

N∑n=0

v(Qn) + h

N−1∑n=0

w

(Qn+1 −Qn

h

)+ a(Q0)− b(QN ) (1)

where h 6= 0 is a parameter of the system v and w are real functions of a real variable. Inorder for the above potential function to satisfy the conditions of Problem Set 2.1 of theprevious section we require w′ to be a surjection and v, w ∈ C1

v : R→ Ωrv w : Ωdw → Ωrw (12)v′ : R→ Ωrv′ w′ : Ωdw → R (13)

with Ωrv′ ,Ωdv ,Ωrv ,Ωdw ,Ωrw ⊆ R and a, b ∈ C1(R).In order to make use of Theorem 2.1 to find the stationary point of the function (1) we

need to write (1) in the form (2). In doing this we have a certain amount of choice: do weinclude v(Qi+1) as part of Ld(Qi, Qi+1) or v(Qi)? We postpone this choice by introducingan extra parameter, β ∈ [0, 1], thus Ld : Ωdw

→ Ωrv∪ Ωrw

,

V (Q0, . . . , QN ) =N−1∑n=0

Lβd (Qi, Qi+1) + hβv(Q0) + h(1− β)v(QN ),

and

Lβd (Qi, Qi+1, h) = hβv(Qi+1) + h(1− β)v(Qi) + hw

(Qi+1 −Qi

h

). (14)

This now enables us to determine the static equilibrium states of the lattice model withpotential energy (1) by searching for solutions to the one of the discrete boundary valueproblems defined in Problem Set 2.2 with

a(Q0) = a(Q0) + hβv(Q0) and (15)b(QN ) = b(QN )− h(1− β)v(QN ). (16)

If we write out the implicit map that results from the definition of the discrete Lagrangiangiven in (14) we get

Pn = −D1Ld(Qn, Qn+1) = −h(1− β)v′(Qn) + w′(Qn+1 −Qn

h

)Pn+1 = D2Ld(Qn, Qn+1) = hβv′(Qn+1) + w′

(Qn+1 −Qn

h

).

2.3 Summary

The stationary points of the potential energy function (1) may be found by solving a discreteboundary value problem. This discrete boundary value problem depends on the behaviourof the two ends of the lattice to be modelled. The possible lattice end behaviours are

5

Q0 QNfixed fixedfixed freefree fixedfree free.

The discrete boundary value problem is given by

Xn =(QnPn

), Xn+1 = φ(Xn) n = 0, . . . , N − 1,

where φ : (Qn, Pn)→ (Qn+1, Pn+1) is defined implicitly by

Pn = −h(1− β)v′(Qn) + w′(Qn+1 −Qn

h

)Pn+1 = hβv′(Qn+1) + w′

(Qn+1 −Qn

h

).

The boundary conditions corresponding to the lattice end behaviour shown above are

Boundary behaviour Q0 P0 QN PN

fixed – fixed α0 free αN freefixed – free α0 free free b′(QN )− h(1− β)v′(QN )free – fixed free a′(Q0) + hβv′(Q0) αN freefree – free free a′(Q0) + hβv′(Q0) free b′(QN )− h(1− β)v′(QN ).

In the above expressions β ∈ [0, 1] is a free parameter which may be chosen freely in thenext section we shall see what effect the parameter β has on the behaviour of the map φand discuss which values are most appropriate.

2.4 Map properties

In this section theorems and results from [9] will be used to demonstrate various properties ofthe iterated map φ that forms the discrete boundary value problems defined in Problem Set2.2. This section and the next gives the explains as to why iterated maps previously derivedin the literature [1, 7] from mechanical lattices have been observed to be area preservingand also related to numerical integrators for certain differential equations.

In order to use the theorems from [9] further assumptions have to be made on the discreteLagrangian Ld(Qn, Qn+1). We have to assume that Ld : Q × Q → R where Qn ∈ Q ⊆ Rand that Ld is invertable with respect to both of its arguments, i.e. the discrete LagrangianLd is regular. In terms of the functions v and w, that form the potential function (1) for anearest neighbour coupled lattice, this requirement is satisfied by requring v, w : R→ R andthat w is invertable. These conditions are more restrictive than those used in Theorem 2.1and so this Theorem still applies.

In the following we need to indicate the dependence of φ on the parameters β and hand so we write φβh to indicate this dependence. In the following list of properties of φβh thebracketed references refer to the appropriate page or theorem of [9].

• The map φ is symplectic, i.e. ψTJψ = J where J =(

0 1−1 0

)and ψ = ∂φ

∂X . The

map φ is the discrete Hamiltonian flow map derived from Lβd and so is symplectic(p. 386).

• The maps φβ and φ1−β for β ∈ [0, 1] are adjoint maps: φ1−β−h φβh = i.d., and so for

β = 1/2 φβh is self adjoint. This is a result of the following property of the discreteLagrangian: Lβd (Qi, Qi+1, h) = −L1−β

d (Qi+1, Qi,−h) (p. 403 Theorem 2.4.1).

• The map φβh is a numerical integrator, with step size h, of the differential equationwith Lagrangian L(q, q) = v(q) + w(q). (Example 2.3.2 p. 402) This relationship isdiscussed further in the next section.

6

These relations show that for the class of lattice systems with potential (1) the staticequilibrium states may be found through the use of a symplectic mapping with a welldefined continuum limit. This is a discrete version of the ‘dynamical phase space analogy’proposed by [6] where by spatial boundary value problems derived from continuous structuralsituations are analysed by first considering the phase space behaviour of the related initialvalue problem.

Having now seen the effect of the parameter β on the iterated map φ one might ask:which value should be chosen? The parameter β could, in pricinpal, take any value in therange [0, 1] but the values 0, 1/2 and 1 are the values that have particular significance. Asdescribed above choosing β = 1/2 means that φ is self-adjoint which may simplify anyproofs on properties of the discrete BVP. This choice of β = 1/2 does, however, increasethe complexity of the boundary conditions for the BVP when modeling free lattice endbehaviour (see (15) and (16)). It may be convenient, especially from a computational pointof view, to choose β = 0 or β = 1 in situations where the behaviour is not the same ateach end of the lattice to simplify the boundary conditions for the discrete boundary valueproblem.

2.5 A link to the continuum problem via geometric integration

The methods of discrete mechanics used in the previous sections allow the definition of a welldefined continuum limit for the discrete mechanical system. After first presenting explicitexpressions for the map φ and the continuum limit just mentioned we see here exactly whattype of numerical integrator the map φ is.

The discrete Lagrangian Lβd (14) is an approximation to an integral

Lβd (Qn, Qn+1, h) =∫ tn+1

tn

v(q) + w(q) dt+O(hr+1)

where q(tn) = Qn, q(tn+1) = Qn+1 and r = 1 for β 6= 1/2 and r = 2 for β = 1/2. Thisintegral is the Lagrangian generating function for the flow of the dynamical system withLagrangian

L(q, q) = v(q) + w(q). (17)

Section 2.3 from [9] then tells us that for small h the map φβh, defined implicitly in ProblemSet 2.2, and given explicitly by φβh : (Qn, Pn)T → (Qn+1, Pn+1)T thus

Qn+1 = Qn + h(w′)−1(Pn + h(1− β)v′(Qn)

)(18a)

Pn+1 = Pn + hβv′(Qn+1) + h(1− β)v′(Qn) (18b)

is a symplectic, order r numerical integrator. This numerical integrator approximates thetime h flow map of the differential equation

w′′(q)q = v′(q). (19)

which is the Euler-Lagrange equation of the Lagrangian (17). We also note at this pointthat this is a Hamiltonian differential equation with Hamiltonian

H(p, q) = p(w′)−1(p)− w ((w′)−1(p))− v′(q)

giving Hamilton’s equations

q =∂H

∂p= (w′)−1(p) p = −∂H

∂q= v′(q).

The above expressions enable us to name the specific type of numerical integrator that themap φβh represents. The symplectic Euler numerical integrator and its adjoint result fromtaking β = 1 and β = 0 respectively [5, p3,43] whilst taking β = 1/2 results in a compositionmethod formed from two steps of length h/2. The second step is a step of the symplectic

7

Figure 2: The mechanical system in its flat state (top) and a displaced state (bottom) from[7] that is studied in Section 3. It consists of a chain of freely pin jointed rigid links of lengthh > 0 constrained to lie in a plane. The vertical springs are linear and the system is axiallyloaded by a force P at its end points.

Euler (SE) method and the first step is a step of the SE’s adjoint method. Alternatively wecan see this method as a partitioned Runge-Kutta method (see [5, p25,p34]) with coefficients

1/2 0 1/21/2 0 1/2

1/2 1/2

1/2 1/2 01/2 1/2 0

1/2 1/2.

3 Example one: Axially loaded linked mechanical struc-ture

In this section we determine the static equilibrium states of the linked mechanical structureshown in Figure 2, and also studied in [7], using the results derived in the previous section.This mechanical structure comprises N freely pin jointed, mass-less, rigid links of lengthh > 0. The displacements Qi, for i = 0, . . . , N , of the joints (mass m) are constrained to liein the same plane and the system is constrained vertically at the ends so that Q0 = QN = 0.The system is also subject to a steady axial load P .

We can write the potential energy of this system V in terms of the energy stored in thesprings U and the work done by P in producing a total end shortening E , where

Ed = h

N−1∑i=0

1−√

1−(Qi+1 −Qi

h

)2 . (20)

After neglecting constant terms V is written

V = U − PE = 12k

N∑i=0

Q2i + Ph

N−1∑i=0

√1−

(Qi+1 −Qi

h

)2

. (21)

We note here this model requires us to work with angular displacements of the lattice’s linksthat satisfy θi ∈ (−π2 , π2 ) and so |Qi+1 −Qi| < h. This energy is now in the standard form(1) with

v(x) = kx2 and w(x) = P√

1− x2.

Restricting the range and domains for these functions and their derivatives to the followingintervals

v : R→ R+ ∪ 0 v′ : R→ Rw : (−1, 1)→ [0, P ] w′ : (−1, 1)→ R

8

means that they satisfy the conditions of Section 2. Therefore we can now use the resultsof this section to immediately derive the discrete boundary value problem that gives theequilibrium states of this mechanical lattice. Theorem 2.1 gives us the boundary valueproblem

Xn+1 = φ(Xn) for n = 0, . . . , N − 1

X0 =(

0W0

), XN =

(0WN

)(22)

where φ : (Qi,Wi)T → (Qi+1,Wi+1)T,

Qi+1 = Qi − h Wi + 2kh(1− β)Qi√P 2 +

(Wi + 2kh(1− β)Qi

)2 (23a)

Wi+1 = Wi + 2kh(βQi + (1− β)Qi+1

), (23b)

W0 and WN are free variables and we have chosen β = 1/2.We can determine the solutions to the discrete BVP (22) numerically (as in [1]) using

a bisection algorithm. Since the solutions to the initial value problem corresponding to theBVP (22) are unique each BVP solution is uniquely parametrised by W0. Choosing aninterval for W0 in which to search and breaking this interval into segments of length ∆W0

we look for sign changes in QN . Once a sign change has been found a suitable zero findingalgorithm can be used to approximate the value of W0 at which QN = 0. We can then, inprinciple, reduce ∆W0 until no further solutions are found. The results of this computationfor N = 6 and N = 16 are shown in Figure 3. Computing the entire solution for each of theseW0 values then allows the computation of the end shortening Ed (see (20)) for particularstatic equilibrium state at a particular load. Solutions are not unique in (end-shortening, p)space but these diagrams help us to determine which solutions we are more likely to observein an experiment. Figure 4 shows these versions of the bifurcation diagrams in Figure 3.

9

00.

51

1.5

01234567

θ 0

p

(a)

Fig

ure

3:B

ifurc

atio

ndi

agra

mfo

rth

est

atic

equi

lbri

umst

ates

ofth

em

echa

nica

llat

tice

show

nin

Fig

ure

2fo

rh

=1

andk

=1

wit

hN

=6

in(a

)an

dN

=16

in(b

).T

hese

exte

ndth

ew

ork

of[7

]to

cons

ider

the

who

lebi

furc

atio

ndi

agra

mfo

rth

ese

finit

ela

ttic

es.

Solu

tion

sto

the

disc

rete

boun

dary

valu

epr

oble

m(2

2)ar

eun

ique

lypa

ram

etri

sed

byθ 0

andP

.

10

00.

20.

40.

60.

81

01234567

E d

p

(a)

Fig

ure

4:T

hese

two

pane

lssh

owth

eso

luti

ons

toth

edi

scre

tebo

unda

ryva

lue

prob

lem

(22)

,fo

rh

=1

andk

=1

wit

hN

=6

in(a

)an

dN

=16

in(b

),as

poin

tsin

load

end-

shor

teni

ngsp

ace

rath

erth

anth

eθ 0p

spac

eus

edin

Fig

ure

3.

11

The results shown in these figures extend the work of [7] to consider the whole bifurcationdiagram for this mechanical lattice with finite size. It also extends the work of [1] todemonstrate that the complex bifurcation diagrams for the static equilibrium states observedtherein are not unique to the mechanical system they study.

There are many complex features shown in Figures 3 and 4. Many of the mechanismsfor the creation of features in complex bifurcation diagrams like this are discussed in [1]for a different mechanical lattice. Indeed it is interesting to compare Figure 3 with Figureof [1]. Also, the thesis [4] contains many results and proofs about the structure of thebifurcation diagrams shown in Figures 3 and 4 which use the discrete boundary value problemformulation developed in this paper.

Many detailed properties of the bifurcation diagrams shown in figures 3 & 4 and thesolutions they depict can be determined by considering the detailed properties of the mapφ. This type of analysis on a different mechanical system can be found in [1] while moredetailed results on the mechanical lattice of this section can be found in [4].

4 Example two: Axially loaded Fermi-Pasta-Ulam lat-tice

In this section we shall use the theory of the Section 2 to to determine, analytically, thestatic equilibrium states of the axially loaded FPU lattice. In this work, by FPU lattice wemean the system with potential energy given by

V (Q) =N−1∑n=0

w(Qn+1 −Qn)− P (Q0 −QN ) (24)

where, as in [2] w(x) = x2/2 + αx3/3 + βx4/4. From Theorem 2.1 we know that, sincewe are considering here free-free boundary conditions, the static equilibrium states of thislattice are given by the discrete boundary value problem:

Qn+1 = Qn + ηn (25a)Pn+1 = Pn (25b)

for n = 0, . . . , N + 1 with ηn ∈ (w′)(−P ). With boundary conditions P0 = PN+1 = −P .In this case, since equation (25a) is so simple we can reduce this BVP to a one dimensionaliterated map

Qn+1 = Qn + ηn

for n = 0, . . . , N with ηn ∈ (w′)(−P ) where (w′)(−P ) denotes the set of solutions, x, to theequation w′(x) = −P . Since this FPU lattice is translationally invariant1 we can arbitrarilypick Q0 = 0. This leaves us with the solution

Qn =n∑n=1

ηn with Q0 = 0. (26)

The number of solutions to this problem now depends on the number of solutions, x, to theequation w(x) = −P . Consider, first, the case P = 0. Solving w′(x) = 0 gives

x = 0 or x =−α±

√α2 − 4β

2β.

This shows that there are three, two and one solution(s) to w(x) = 0 for 4β > α2 , 4β = α2

and 4β < α2 respectively. Solving w′′(x) = 0 then shows that w′(x) is strictly monotonicallyincreasing for 3β > α2. This situation is summarised in Figure 5.

The number of solutions to equation (26) can now be determined, and hence also thenumber of static equilibrium states to the unloaded (P = 0) FPU lattice. With only oneelement in the set (w′)(0) there is only one solution to (26), when this set contains twoelements there are 2N different solutions and with three elements in (w′)(0) there are 3N

different static equilibrium states.1i.e. If we let Qn → Qn + γ for a fixed constant γ the potential (24) does not change in value.

12

−4 −2 0−1

−0.5

0

0.5

1

x

w′ (

x)

−4 −2 0−1

−0.5

0

0.5

1

x−4 −2 0

−1

−0.5

0

0.5

1

x−4 −2 0

−1

−0.5

0

0.5

1

x(a) (b) (c) (d)

Figure 5: Plots of w′(x) for α = 1 and β = 0.4 (a), β = 0.3 (b), β = 0.25 (c) and β = 0.2(d). This illustrates the behaviour of w′(x) as we change α and β. For 3β > α2 w′(x) ismonotonically increasing (a), at 3β = α2 w′(x) we loose monotonicity (b), at 4β = α2 w′(x)the number of solutions to w′(x) = 0 changes to two (c) and for 4β < α2 w′(x) there arethree solutions to w′(x) = 0 (d).

4.1 The effect of load

The final question we answer in this section is: what effect does the axial load have on thenumber of static equilibrium states in the FPU lattice? The answer to this can be found byconsidering the fact that ηn ∈ (w′)(−P ) and looking at Figure 5. The number of solutionsto w′(x) = −P is found by considering the intersection of a horizontal line at height −Pand the plot of w′(x). We can see that for α2 < β the load has no effect on the number ofsolutions as w′(x) is monotone. For α2 < β < β, although the unloaded lattice has only oneequilibrium state, by compressing the lattice we can pass into the parameter regions with2N and 3N equilibrium states. For β > 3 the unloaded lattice does contain spontaneousstatic equilibrium states, but interestingly, we can remove these by either putting the latticeunder extra compression or tension.

5 Conclusions

In this paper it was shown that it is possible to determine the static equilibiurm states of ageneral nearest-neighbour coupled lattice (with potential energy function given by (1)) byusing a discrete boundary value problem (DBVP). This DBVP, summarised in Section 2.3allows the modelling of all possible behaviours at each end of the lattice. That is fixed orfree displacement coordinates at the lattice end points. Further, in Section 2.4, the closenessof these results to the theory of discrete mechanics (see for example [9]), allowed the detaileddetermination of the properties of the iterated map that formed the discrete boundary valueproblem.

Section 2.5 used information from the geometric integration literature to show that,under certain assumptions, each nearest-neighbour coupled lattice model and correspondingdiscrete boundary value problem has a related continuous boundary value problem. In fact,the iterated map that forms the discrete boundary value problem is a geometric numericalintegrator for the differential equation that forms the continuous boundary value problem.

Section 3 saw the results of Section 2 put into practice and used to determine, numerically,the static equilibrium states of a pin-jointed mechanical structure. These results extendedthe previous work of [7] to determine the full bifurcation diagram for these static equilibriumstates.

Finally, in Section 4 the results of Section 2 were applied to the Fermi-Pasta-Ulam latticewith its two parameters α and β. It was shown, analytically, that for α2 < 4β there is onlyone static equilibrium state for the unloaded FPU lattice but for α2 = 4β this increases to2N while for α2 > 4β there are 3N static equilibrium states. Moreover, for α2 > 3β thenumber of static equilibrium states of the lattice can be moved between the three values 1,2N and 3N by changing the axial load on the lattice.

13

References

[1] G Domokos and P Holmes. Euler’s problem, Euler’s method, and the standard map; or,the discrete charm of buckling. Journal of Nonlinear Science, 3:109–151, 1993.

[2] E Fermi, J Pasta, and S Ulam. Studies of non linear problems. 1955. Los AlamosReport LA-1940 since published in ‘The Collected Papers of Enrico Fermi, Vol II (USA1939-1954)’, University of Chicago Press, 1965, p978.

[3] S Flach and C R Willis. Discrete breathers. Physics Reports, 295:181–264, 1998.

[4] S C Green. The statics and dynamics of mechanical lattices. PhD thesis, in preparation,University of Bath.

[5] E Hairer, C Lubich, and G Wanner. Geometric Numerical Integration. Springer Seriesin Computational Mathematics. Springer, 2002.

[6] G W Hunt, H M Bolt, and J M T Thompson. Structural localization phenomena andthe dynamical phase-space analogy. Proceedings of the Royal Society of London. A.Mathematical and Physical Sciences, 425:245–267, 1989.

[7] G W Hunt, R Lawther, and P Providencia E Costa. Finite element modelling of spa-tially chaotic structures. International Journal for Numerical Methods in Engineering,40:2237–2256, 1997.

[8] Attila Kocsis and Gyorgy Kaarolyi. Conservative spatial chaos of buckled elastic linkages.Chaos, 16(3):033111, 2006.

[9] J E Marsden and M West. Discrete mechanics and variational integrators. Acta Numer-ica, 10(-1):357–514, 2001.

14