Discrete Invariant Variational Problems

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Master of Science in Physics and Mathematics

June 2011

Brynjulf Owren, MATH Submission date:

Supervisor:

Norwegian University of Science and Technology Department of Mathematical Sciences

Discrete Invariant Variational Problems

Geir Bogfjellmo

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Problem Description

Study how moving frames can be used to develop numerical methods for variational problems such that the numerical method inherit symmetries from the continuous problem.

Assignment given: 24 January 2011.

Supervisor: Brynjulf Owren

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ABSTRACT

This thesis studies variational problems invariant under a Lie group transformation, and invariant discretizations of these. In chapters two and three, a general method for creating symplectic integrators preserving certain classes of variational symmetries of first order Lagrangians is developed and demonstrated. In chapters four and five, it is assumed that the discrete Lagrangian is invariant under a certain group action, and the Euler–Lagrange equations for the variational problem are expressed in the invariants of the group action.

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PREFACE

This thesis was written for the course TMA4910 – Numerical Mathematics, Master Thesis in the spring term of 2011. It is the final part of my Master’s degree in Applied Physics and Mathematics, with Specialization Industrial Mathematics.

The thesis is to a large degree a continuation of my specialization project on symmetry-preserving numerical methods. After studying the techniques for in- variantizing numerical schemes, it seemed natural to continue on to variational problems, since variational symmetries are closely connected to conservation laws.

The writing of the thesis would not have been possible without the support of several people. I would like to thank my supervisor Brynjulf Owren, for encourage- ment, guidance and proof reading. I would further like to thank Elena Celledoni, Tore Halvorsen and Olivier Verdier for additional help and proof reading.

Finally I would like to thank all my friends in Matteland, for countless coffee-, lunch-, dinner- and football breaks, occasionally at bizarre hours.

Geir Bogfjellmo,June 27, 2011

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CHAPTER 1 INTRODUCTION

When studying an object in mathematics or other fields, symmetries, transforma- tions which do not change the object, are often useful for simplifying or reducing a problem. The symmetries of our concern are Lie group actions acting on a manifold. The machinery of moving frames provides tools for calculating and studying objects with such symmetries. We use the moving frame formalism as developed by Fels and Olver [4, 3, 14]. An elementary introduction, which also provides material on the variational problems studied in this thesis is the book by Mansfield [8].

Our main goal is to study variational problems and discretizations of these.

The objects to be studied are functions f : X →U, where X and U are smooth manifolds. While the theory in this thesis makes no further assumptions onX and U, the numerical examples will haveX=R,U =Rⁿ.

The thesis assumes the reader to be familiar with basic differential topology concepts including the concept of manifolds, tangent bundles and tangent maps.

It also assumes the reader is familiar with the basic concepts of Lie groups and Lie algebras.

A few notes on assumptions and notations

• The Einstein summation convention is used. If an index appears twice in a term it is an indication that this index should be summed over.

• Function and manifolds are assumed to be C^∞ smooth, unless otherwise noted.

• We use the two-argument atan2 in some formulas, to avoid the usual quadrant issues of arctan. Mathematically, atan2(y, x) is the principal argument of the complex numberx+iy.

• Mostly, low indices are used for elements of a sequence, typically a numerical solution, while high indices are used for coordinates, though his convention has not been followed completely.

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CHAPTER 2 GROUP ACTIONS AND MOVING FRAMES

2.1 Basic Definitions and Results

Definition 1. Let Gbe a Lie group with identity e and M a smooth manifold.

Furthermore, let U ⊂G×M be an open set with {e} ×M ⊂U. A smoothlocal left group action is a smooth function Ψ :U →M satisfying

(a) If (h, z)∈U, (g,Ψ(h, z))∈U and (gh, z)∈U, then Ψ(g,Ψ(h, z)) = Ψ(gh, z).

(b) For allz∈M,

Ψ(e, z) =z (c) If (g, z)∈U, then also (g⁻¹,Ψ(g, z))∈U and

Ψ(g⁻¹,Ψ(g, z)) =z

If U =G×M, we say Ψ is aglobal left group action. In this case, part (c) above follows from parts (a) and (b).

Alocal right group action satisfies part (b) and (c) above and (a’) If (h, z)∈U, (g,Ψ(h, z))∈U and (hg, z)∈U, then

Ψ(g,Ψ(h, z)) = Ψ(hg, z).

We will restrict the study to connected group actions, which means that in addition to (a), (b) and (c),

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(d) GandM are connected as manifolds, (e) U is connected,

(f) Gz={g∈G|(g, z)∈U}is connected for everyz∈M. We will make use of the alternative notations

g·z= Ψ(g, z) if the action is left, and

z·g= Ψ(g, z)

if the action is right. In local coordinatesz= (z¹, . . . , z^m) on the manifold, we will sometimes abuse this notation and write e.g.g·zⁱ for theith coordinate ofg·z.

Additionally we will use the notations Ψg(z) =g·z Ψz(g) =g·z for the group action with either argument fixed.

The difference between local and global group actions is somewhat technical. In this thesis definitions are mostly stated just for the global case. The definitions for local group actions are analogous, but with additional restrictions. Additionally, most group actions of our concern are left.

For theoretical purposes it is sometimes useful to restrict the action to a connected open set V ∈ M, which requires that the domain ofU is restricted such that all the axioms above hold.

We will be interested in functions that are invariant or equivariant under a left group action.

Definition 2.

(a) A function f :M →Risinvariant under the group action if f(g·x) =f(x)

for all (g, x)∈G×M.

(b) Given two manifoldsM andN, and group actions Ψ^M :G×M →M

Ψ^N :G×N →N

with common Lie group G, a function F : M → N is equivariant if for all g∈G

F◦Ψ^M_g = Ψ^N_g ◦F as functionsM →N

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2.1. BASIC DEFINITIONS AND RESULTS 5 The right-equivariant moving frames are equivariant maps fromM to G, with the left action ofGon itself defined by Ψ_g(h) =hg⁻¹.

Definition 3. A right-equivariant moving frame for a group action is a smooth functionρ:M →Gsatisfying the right-equivariance

ρ(g·z) =ρ(z)·g⁻¹ (2.1)

for allg∈G,z∈M.

Remark: Ifz7→ρ(z) is a right-equivariant moving frame,z7→ρ(z) =e ρ(z)⁻¹is a left-equivariant moving frame, satisfyingρ(ge ·z) =g·ρ(z). While the classic movinge frames pioneered by Cartan [1] and others, are left-equivariant, Fels and Olver [4]

introduced the right-equivariant version as more practical for computations, and in this thesis, all moving frames are right-equivariant.

Under some assumptions on the group action, a local moving frame, defined in a neighbourhood of an arbitrary point z∈M, exists, and can be constructed. To state these assumptions, we need some terminology.

Definition 4.

(a) For a pointz∈M the(group) orbit is the submanifoldO(z) ={g·z|g∈G}

(b) The isotropy subgroupof a pointz∈M isG_z={g∈G|g·z=z}.

(c) A group action is:

• Locally freeifG_z is a discrete subgroup for allz∈M.

• FreeifGz=efor allz∈M.

• Regularif all orbits are of the same dimension, and for each pointx∈M, there exists a neighbourhood U of x such that the intersection between an orbit andU is either empty or connected.

• Locally effectiveif theglobal isotropy subgroupG^?_M =T

z∈MG_zis discrete.

• EffectiveifG^?_M =e.

(d) A submanifold which intersects each group orbit transversally and exactly once, is across-section.

We note that the adjective “locally” is natural in the sense that if Ψ is locally free, (resp. effective), then by suitably restricting the domain of Ψ, the resulting local group action is free (resp. effective). If the group action is free and regular, then for any pointz∈M there is a neighbourhoodU ⊂M, such that there exists a cross-sectionK ⊂U for the group action restricted toU.

Theorem 1. Let G act freely and regularly on M, and let K be a cross-section.

Given z ∈ M, let ρ(z) ∈ G be the unique group element such that ρ(z)·z ∈ K.

Thenρ:M →Gis a (right-equivariant) moving frame.

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Proof. Letz∈U, andh∈Gsuch thath·z∈U. Since each group orbit intersects Kexactly once,ρ(h·z)·h·z=ρ(z)·z. Since the action is free,ρ(h·z)·h·ρ(z)⁻¹=e and the equivariance follows.

Usually, the cross-section is defined in local coordinatesz= (z¹, . . . , z^m) as the locus of a set of equations

Zⁱ(z) = 0, i= 1, . . . , r,

and the moving frame can be obtained by solving the equations Zⁱ(ρ(z)·z), i= 1, . . . , r,

with respect to the group elementρ(z).

The choice of cross-section K and moving frame induces an invariantization operator on the space of functions onM. IfF is any function onM, its invarianti- zationι(F) is invariant under the group action and is defined by

ι(F)(z) =F(ρ(z)·z).

Of special importance are the invariantizations of the coordinate functionsz7→zⁱ. These are the fundamental invariants Iⁱ=ι(zⁱ). In coordinates, the invariantization of a function is

ι[F(z¹, . . . , z^m)] =F(I¹(z), . . . , I^m(z)).

Since the invariantization of an invariant function is the function itself, it follows that any invariant function can be written in terms of the fundamental invariants.

This result, known as the replacement theorem, is often useful.

Theorem 2 (Replacement theorem). If F is an invariant function, then F(z¹, . . . , z^m) =ι[F(z¹, . . . , z^m)] =F(I¹(z), . . . , I^m(z)).

In particular, the invariantizations of the cross-section equationsι Zⁱ

are con- stant. These are thephantom invariants.

2.2 Infinitesimals

Let g = TeG be the Lie algebra of the Lie group G. The tangent map at the identity

TeΨz:g→TzM defines the infinitesimal action

ψ:g×M →T M.

Ifv1, . . . , vr form a basis for the Lie algebra, the corresponding vector fields vj(z) =ψ(vj, z) =ψ_i^j(z)∂_zi

are the generatorsof the group action. The basis will typically be defined by local coordinates (g¹, . . . , g^r) onGnear the identity such thatvj(z) =_∂g^∂jΨ(g, z)

_e.

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2.3. PROLONGATION 7

2.3 Prolongation

For any action Ψ on a manifoldM, there are induced actions onM’s tangent bundle and more generally, thejet bundlesJⁿ(M, p) overM. Jⁿ(M, p) is defined as the set of equivalence classes ofp-dimensional submanifolds under the equivalence relation ofnth order contact at a single point. The induced action on a jet bundle is known as the prolongation of the action. For the tangent bundle, the prolonged action

Ψ^{T M} :G×T M →T M is defined by

Ψ^{T M}_g =TΨg

that is, differentiation with respect to the manifold variable. The actions on higher order jet bundles are defined in a similar manner.

Prolongation has a regularizing effect, if Ψ acts effectively on open subsets, meaning that the only group element fixing every point in an open subset ise, then for a sufficiently largen, the prolongation of Ψ acts locally freely and regularly on an open dense subset ofJⁿ(M, p) [13], so that even if Ψ does not admit a moving frame, Ψ^Jⁿ^(M,p) fornsufficiently large does.

Locally onJⁿ(M, p), we can splitM intoX×U, whereXisp-dimensional and considerp-dimensional submanifolds as smooth functionsf :X→U. For indexing variables and invariants, it is convenient to usemulti-indices. With (x¹, . . . , x^p) local coordinates onX, (u¹, . . . , u^m−p) local coordinates onUwe useK= (k₁, . . . , k_l) to index the partial derivatives

u^α_K = ∂^lu^α

∂x^k¹· · ·∂x^k^l,

and use the (xⁱ, u^α_K) as local coordinates onJⁿ(M, p) =X×U⁽ⁿ⁾. The corresponding fundamental invariants areI_K^α =ι(u^α_K).

We also define the differential operator D_K = ∂^l

∂x^k¹· · ·∂x^k^l.

Due to the replacement theorem, any invariant expressed inu^αand derivatives of these can be written in terms of the fundamental invariants. For simpler notation we will often write simply Ψ also for the prolonged action of Ψ, and use the multi- indices to index the elements of Ψ(g, z), wherez is an element of the jet bundle.

The generating vector fields on Jⁿ(M, p) are given in terms of the original generators by the prolongation formula. In local coordinates, if the original vector field is

v=ξⁱ∂_xi+φ^α∂_uα

the prolonged vector field is

pr v=v+X

α,K

φ^α_K∂_u^α

K

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where φ^α_K =D_K φ^α−ξⁱu^α_i

+ξⁱu^α_K,i.

A group action onM also induces naturally an action onMⁿ=M × · · · ×M Ψ^M_g ⁿ(z1, . . . , zn) = (Ψg(z1), . . . ,Ψg(zn)).

which we will call the prolongation to Mⁿ. If v(z) is a generator of the original group action Ψ, the corresponding generator for Ψ^Mⁿ is

v^Mⁿ=

n

X

i=1

vi(zi) where v_i(z_i)∈T_z_iM_i.

2.4 Differentiation of Invariants

We mainly consider cases where the manifold can be split intoX×U wherex∈X represent the independent variables andu∈U the dependent variables. Further- more, we assume thatxis invariant under the group action. It is important to note that invariantization and differentiation do not commute, even if the differentiation is with respect to an invariant variable. However, it is possible to calculate derivatives of the fundamental invariants I_K^α in terms of other fundamental invariants.

Combined with the replacement theorem, this can be used to differentiate any invariant function. Fels and Olver [3, Section 13] showed that these expressions can in fact be calculated from the prolonged vector fields and the equations for the cross-section. What follows is a proof of these relations for the case of invariant independent variables.

Lemma 1. Define Rg:G→Gby Rg(h) =hg. Then TeΨg·z=TgΨz◦TeRg

=TgΨz◦(TgR_g−1)⁻¹

Proof. The first equality follows from the identity Ψz◦Rg= Ψg·z

and the chain rule. The second equality follows fromT(Rg◦R_g−1) =id.

Lemma 2.

T_zρ=T_ρ(g·z)R_g◦T_g·zρ◦T_zΨ_g for all g∈G, and specifically, forg=ρ(z)

Tzρ=TeR_ρ(z)◦T_ρ(z)·zρ◦TzΨ_ρ(z)

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2.4. DIFFERENTIATION OF INVARIANTS 9 Proof. ρ(z) =ρ(g·z)·g for all z, thusρ=R_g◦ρ◦Ψ_g. The lemma follows from the chain rule.

Theorem 3. Assume thatΨ :G×M →M has a cross-sectionKwith correspond- ing moving frame ρ:M →G. Let z= (z¹, . . . , z^m) be local coordinates near some point x∈ Kand assume that in these coordinatesK is defined as the kernel of the function

Z(z) = (Z¹(z), . . . , Z^r(z))∈R^r.

Furthermore letv1, . . . ,vk,where thevj =ψ_i^j(z)∂_zⁱ be the generators of the action.

Define the matrices

J(z) =T_ρ(z)·zZ, ψ(z) =T_eΨ_ρ(z)·z with entries

J(z)i,j =∂Zⁱ

∂y^j y=ρ(z)·z

ψ(z)i,j =ψ_i^j(ρ(z)·z), The tangent map of the invariantization map

ι:z7→Ψ(ρ(z), z) is in local coordinates given by

T_zι=T_zΨρ(z)−ψ(J ψ)⁻¹JT_zΨρ(z).

Proof. By the product rule and lemmas 1 and 2 Tzι=T_ρ(z)Ψz◦Tzρ+TzΨ_ρ(z)

=T_eΨ_ρ(z)·z◦(T_eR_ρ(z))⁻¹◦T_eR_ρ(z)◦T_ρ(z)·zρ◦T_zΨ_ρ(z)+T_zΨ_ρ(z)

=TeΨ_ρ(z)·z◦T_ρ(z)·zρ◦TzΨρ(z)+TzΨρ(z)

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Assume thatZ¹, . . . , Z^k, z^k+1, . . . , z^mare functionally independent nearx(if not, rearrange thezⁱ), so thatη(z) = (Z¹, . . . , Z^k, z^k+1, . . . , z^m) form local coordinates.

The Jacobian of the transformation mapη, at ρ(z)·z∈K is T_ρ(z)·zη=

J A

,

where J is as defined above, and Aplays no further role.

By construction,Z =π◦η, whereπis projection onto the firstkcoordinates.

By ρ(z)·z∈K, we have

Z◦ι=π◦η◦ι= 0

Taking the tangent map ofπ◦η◦ιat the cross-section gives T_η(ρ(z)·z)π◦T_ρ(z)·zη◦T_ρ(z)·zι= 0.

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Since

T π= id 0

, we deduce that

T_ρ(z)·zη◦T_ρ(z)·zι= 0

A

, (2.3)

for someA(different from the previousA). The topkrows of the matrix equation (2.3) are

J ◦T_ρ(z)·zι=0.

Inserting from equation (2.2), we get

J(TeΨ_ρ(z)·z◦T_ρ(z)·zρ◦TzΨe+TzΨe) =0 J ψ◦T_ρ(z)·zρ=−J,

where the last line is due to T_zΨ_e = id. Since the cross-section is assumed to intersect the orbits of the group action transversally,J ψ is of full rankr, so this implies

T_ρ(z)·zρ=−(J ψ)⁻¹J. Inserting this into equation (2.2) completes the proof.

In our applications, the Zⁱ depend on only a few of thezⁱ, say ζ¹, . . . , ζ^l. Let J⁰ be the non-zero columns ofJ

J_i,j⁰ = ∂Zⁱ

∂ζ^j.

We apply the theorem to fundamental invariants of a prolonged action. We let the manifold be Jⁿ(M, p), which we split as before and use local coordinates z= (u¹, . . . , u¹₁, . . . , u^α_K, . . .). Where the indices on matrices in theorem 3 refers to coordinatezⁱ, we instead index with (α, K).

For a fundamental invariantI_K^α =ι(u^α_K) =ι(z)α,K, which we differentiate with respect to the invariant variablexⁱ, we have by the theorem

∂

∂xⁱI_K^α = ∂

∂xⁱ (ι(z)_α,K)

= ∂

∂xⁱι(z)

α,K

=

Tzι ∂z

∂xⁱ

α,K

=

T_zΨ_ρ(z) ∂z

∂xⁱ

α,K

−

ψ(J ψ)⁻¹JT_zΨ_ρ(z) ∂z

∂xⁱ

α,K

=

TzΨ_ρ(z) ∂z

∂xⁱ

α,K

−ψ_α,K(J ψ)⁻¹JTzΨ_ρ(z)∂z

∂xⁱ,

where the subscripts on matrices refers to rows. The first term of the final right hand side is equal to ι(u^α_Ki) =I_Ki^α by the definition of prolongation. The second

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2.4. DIFFERENTIATION OF INVARIANTS 11 term can be simplified by deleting the zero columns ofJand the corresponding rows ofψ and elements ofT_zΨ_ρ(z)_∂x^∂_iz. The remaining rows ofψare the coefficients of

∂_ζj in the generators, evaluated at the cross-section, and the remaining elements in TzΨ_ρ(z)_∂x^∂iz are the fundamental invariants ι_∂ζj

∂xⁱ

, again by the definition of prolongation.

Letψ^ζ be the remaining rows ofψ, and let Ti,j =ι

∂ζⁱ

∂x^j

. and define thecorrection matrix

C =−(J⁰ψ^ζ)⁻¹J⁰T.

The correction matrix, which only depends on the ζⁱ appearing in the cross- section equations and their first derivatives, provide the correction terms relating invariantization and derivation. We sum up the discussion above in a theorem.

Theorem 4. The derivatives of the fundamental invariants are

∂_xiI_K^α =I_Ki^α +C_j,iψ_α,K^j , (2.4) whereψ_α,K^j = ^∂u_∂g^α^K

j

_g=e andC is as described above.

The symbolic differential formulas above gives a way to express certainI_K^α as functions of lower-order invariants and derivatives of these. We will call a finite set of invariantsIgensuch that all invariants can be written as a function of invariants inIgenand a finite number of their derivatives for aset of generators. It is a classic result that a finite set of generators always exists. A more recent result is that if the frame isminimal, then the set

ι(x^j), ι(u^α), ι ∂Zⁱ

∂x^j

is a set of generators. [7]

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CHAPTER 3 MECHANICAL SYSTEMS

We approach classical mechanics as formulated by Lagrange and Hamilton from a geometric point of view. Important sources for the theoretical background in this chapter are a book by Marsden and Ratiu [9], and an article by Marsden and West [10]. We follow these sources and consider Hamiltonian mechanics as vector fields on symplectic manifolds.

3.1 Symplectic Manifolds and Symplectic Maps

We first recall some concepts from differential topology.

Definition 5. LetM be a smooth manifold.

• Adifferentialk-formω overM assigns to any pointq∈M ak-linear, alter- nating map

ω_q :T_qM× · · · ×T_qM →R,

in a smooth manner. We identify differential zero-forms with smooth func- tionsM →R.

• Thecotangent bundleoverM,T^∗M is the set of elements of the form (q, p), whereq∈M and

p:T_qM →R

is linear. Differential one-forms overM are sections ofT^∗M.

• Thewedge product of a k-form ω and an l-formξ, is thek+l form defined 13

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by

(ω∧ξ)q(v1, . . . vk+l)

=(k+l)!

k!l!

X

π

(sgnπ)ω_q v_π(1), . . . , v_π(k)

ξ_q v_π(k+1), . . . , v_π(k+l) where all the vi lie in TqM and the sum is over all permutations of k+l elements. sgnπis the sign of the permutation,

sgnπ=

(1 ifπis even

−1 ifπis odd.

• Ifvis a vector field onM, theinterior derivativeivis a map fromk+ 1-forms tok-forms defined by

(ivω)_q(w1, . . . , wk) =ωq(v|q, w1, . . . , wk)

where all thew_i lie inT_pM andv|q is the vector field evaluated atq.

• Theexterior derivative d is a map fromk-forms to (k+ 1)-forms satisfying (a) Iff is a 0-form, i.e. a function. df is the normal tangent map.

(b) d is linear.

(c) d satisfies the product rule. Ifω is a k-form andξanl-form then d(ω∧ξ) = dω∧ξ+ (−1)^kω∧dξ

(d) d(dω) = 0 for allω.

• IfF :M →N is a smooth function between manifolds, itspull-backF^∗maps differential forms overN to differential forms overM. Letωbe a differential k-form overN, andq∈M, andv1, . . . , vk ∈TqM.

(F^∗ω)q(v1, . . . , vk) =ω(TqF v1, . . . , TqF vk).

Pull-backs of functions commute with the wedge product and the exterior derivative [9, Section 4.2].

• Ifva vector field onM andφtits flow, then theLie derivativeof ak-form is Lvω= ∂

∂tφ^∗_tω _t=0

• IfF:M →N is a diffeomorphism, itscotangent lift T^∗F is a map T^∗F :T^∗N →T^∗M

such that

T^∗F(q, p) = (F⁻¹(q),p)˜ where ˜pis the linear function onT_F−1(q)M defined by

˜

p(v) =p(T_F⁻¹_(q)v)

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3.1. SYMPLECTIC MANIFOLDS AND SYMPLECTIC MAPS 15 We will also needCartan’s magic formula[9, Theorem 4.3.3].

Lvω= d(ivω) +iv(dω).

Asymplectic manifold is a smooth manifoldM, equipped with a nowhere van- ishing two-form Ω. The two-form relates a smooth HamiltonianH :M →Rto the Hamiltonian vector fieldv_H defined by the relation

iv_HΩ = dH (3.1)

The integral curves ofvHgenerate the flow of the Hamiltonianφt. The flow satisfies the two properties

• φ^∗_tH =H

• φ^∗_tΩ = Ω Proof. Indeed,

∂

∂tφ^∗_tH

_t=0=LvHH

= dH(v_H)

= Ω(vH,vH)

= 0

∂

∂tφ^∗_tΩ

_t=0=Lv_HΩ

=iv_HdΩ + d(iv_HΩ)

= 0 + d²H

= 0

The last property above implies that the symplectic form is preserved under the pull-back of φt for all t. Maps with this property are called symplectic maps. As this is an important property of Hamiltonian flows, it seems natural to search for numerical schemes which also have this property. Such numerical schemes are known assymplectic integrators, and have been widely studied.

In most practical applications, the symplectic manifoldM arises as the cotangent bundle over some configuration manifoldQ. We write points in T^∗Qas (q, p) and writehp, vifor pairing of covectors and vectors overQ, andω·uto denote the pairing of covectors and vectors overM =T^∗Q. In both cases, the base point of the covector and vector is assumed to be the same. Additionally, we will sometimes abuse notation and writeh(q, p), viforhp, viwith base point q.

When M = T^∗Q, the symplectic form is the canonical two-form, defined as follows. Let Θ be the canonical one-form onT^∗Q. Ifuis a vector in T_(q,p)(T^∗Q) then

Θ(q, p)·u=

p, T_(q,p)π u

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where π : T^∗Q → Q is the natural projection. The canonical two-form is then Ω =−dΘ.

MapsF :T^∗Q→T^∗S such that

F^∗ΘS = ΘQ,

where ΘQ and ΘS are the canonical one-forms on their respective cotangent bundles, are special symplectic maps. The special symplectic maps F : T^∗Q →T^∗S are exactly the cotangent lifts of diffeomorphismsf :S→Q[9, Proposition 6.3.2].

If

q= (q¹, . . . , qⁿ) are local coordinates on Q, we can expand with

p= (p¹, . . . , pⁿ)

to get local coordinates on T^∗Q. We will choose the pⁱ to correspond with the natural basis onT Q, i.e. ifv=vⁱ∂_qi ∈TqQ, then

hp, vi=pⁱvⁱ. In such coordinates

Θ =pⁱdqⁱ Ω = dqⁱ∧dpⁱ,

where dqⁱ and dpⁱ are the exterior derivatives of the coordinate functionsqⁱandpⁱ The following theorem, whose proof can be found in [10, Sections 1.4.4-5], shows that the canonical one-form Θ determines symplectic maps.

Theorem 5. Let

F :T^∗Q→T^∗Q

be a smooth map and letΓ(F)⊂T^∗Q×T^∗Qbe its graph. Furthermore let π_1,2:T^∗Q×T^∗Q→T^∗Q

be the projections onto each of the components, i: Γ(F)→T^∗Q×T^∗Q the inclusion map, and

Θ =ˆ π^∗₂Θ−π^∗₁Θ.

ThenF is symplectic if and only if there exists, at least locally, a smooth function S: Γ(F)→Rsuch thati^∗Θ = dS. The functionˆ S is called thegenerating function of F, and we say thatF is generated byS.

A special case occurs if (π1×π2)◦i: Γ(F)→Q×Qis a diffeomorphism. In this case, S can be written as a functionS :Q×Q→Rand is a generating function of the first kind. In coordinates, the relation between F andS becomes

F: (q1, p1)7→(q2, p2)

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3.2. LAGRANGIAN AND HAMILTONIAN SYSTEMS 17 where

p₁=−D1S(q₁, q₂) p2=D2S(q1, q2)

and D1 and D2 are the partial differential operators with respect to q1 and q2, respectively.

Generating functions of the first kind have a nice interpretation when combined with Hamilton’s principle. To arrive at this interpretation and its consequences for symmetries and discrete mechanics, we first need to recall some classical mechanics.

3.2 Lagrangian and Hamiltonian Systems

In Lagrangian mechanics, one considers mechanical systems on some configuration spaceQ, for which one can define kinetic energyT(q,q) and potential energy˙ U(q).

The mechanical system follows a path in Q which minimizes the integral of the LagrangianLdt= (T−U)dt. This isHamilton’s principle. By variational calculus, it can be shown that such paths satisfy the Euler–Lagrange equations, which in local coordinates (q¹, . . . , qⁿ) are

Eⁱ(L) =−d dt

∂L

∂q˙ⁱ

+∂L

∂qⁱ = 0.

In most of the interesting cases, the Lagrangian formalism is equivalent to the Hamiltonian formalism. In Hamiltonian mechanics, systems are defined by their Hamiltonian function defined on the cotangent bundle of the configuration space.

The system follows integral curves of the Hamiltonian vector field, which is defined by (3.1), or in coordinates

v_H= ∂H

∂pⁱ

∂

∂qⁱ −∂H

∂qⁱ

∂

∂pⁱ

.

The relation between the Lagrangian and Hamiltonian formulation is theLeg- endre transformation F L:T Q→T^∗Q,defined by

F L(q, v) = (q, p) such that

hp, wi= ∂

∂L(q, v+w)

₌₀. (3.2)

Or in coordinates pⁱ= _∂^∂L_q_˙i.

The Hamiltonian corresponding to the variational problem is then

H(q, p) =hp, vi −L(q, v) (3.3)

We assume that the Lagrangian ishyperregular, which means that the Legendre transform is a diffeomorphism with inverseF H :T^∗Q→T Qgiven by

F H(q, p) = (q, v) such that

hα, vi= ∂

∂H(q, p+α)

_=0. (3.4)

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3.3 Symmetries

We first explore how invariance under a group action relates to the Lagrangian and Hamiltonian formulations. The cotangent lift enables us to prolong a left Lie group action Ψ :G×Q→Qto a right action on Ψ^T^∗^Q:G×T^∗Q→T^∗Q, by

Ψ^T_g^∗^Q =T^∗Ψg.

The generating vector fields on the cotangent bundle are defined bycotangent lift. For a generating vector fieldv onQ, let v∈gbe the corresponding element in the Lie algebra. Then its cotangent lift

v^T^∗^Q = ∂

∂T^∗Ψg

₌₀, where gis a path in in Gwithg₀=eand ^∂g_∂

₌₀=v.

In coordinates, ifv=ψⁱ∂qⁱ, then the cotangent lift is v^T^∗^Q=−ψⁱ∂_qi+p^j∂ψ^j

∂qⁱ∂_pi.

This formula is equivalent to the formula in [10, Section 1.4.2], but with terms simplified. We include a proof.

Proof. Let F = Ψg and T^∗F(q, p) = (˜q,p), and use coordinates (q˜ ⁱ, pⁱ), (˜qⁱ,p˜ⁱ) from the same coordinate chart. From the definition of cotangent lift of the action,

˜

q=F⁻¹(q). So for the coefficients of the vector field corresponding to theqⁱ,

∂q˜ⁱ

∂

₌₀= ∂

∂ F⁻¹(q)i

₌₀=−ψⁱ.

For the remaining terms, we let w=wⁱ∂_q_˜i be an arbitrary vector with base point

˜

q. We writeh˜p, wi= ˜pⁱwⁱ, and similar forpand use the definition of cotangent lift to get

h˜p, wi=hp, Tq˜Fwi

˜

pⁱwⁱ=p^j∂(F(˜q))^j

∂q˜ⁱ wⁱ

Differentiating the individual ˜pⁱwith respect togives the coefficients corresponding to thepⁱ in the lifted vector field.

∂p˜ⁱ

∂

₌₀=p^j∂²(F(˜q))^j

∂q˜ⁱ ₌₀

=p^j∂ψ^j

∂q˜ⁱ ₌₀

Where the ˜q in the last line can be replaced by q since F₀ = Ψe is the identity map.

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3.3. SYMMETRIES 19 A LagrangianLis invariant under the prolongation of a group action Ψ if

L(q, v) =L(Ψ_g(q), T_qΨg(v)) (3.5) for allg∈Gand (q, v)∈T Q.

Theorem 6. Let the Legendre transform F L be defined as in (3.2) and assume that L is invariant under the prolongation of the group action Ψ : G×Q → Q.

Then the following diagram commutes.

T Q

TΨ_g

F L //T^∗Q

T Q F L //T^∗Q

T^∗Ψ_g

OO

Proof. Let (q, v)∈T Qandg∈Gbe arbitrary. We calculate

hT^∗Ψ_g◦F L◦TΨ_g(q, v), wi=h(F L◦TΨ_g(q, v)), TΨ_g(q, w)i

= ∂

∂L(Ψ_g(q), T_qΨ_g(v) +T_qΨ_g(w))

= ∂

∂L(q, v+w)

where the last equality is due to the linearity ofTqΨg and the invariance ofL.

Theorem 7. Let H be defined as in (3.3). H is invariant under the pull-back of the action Ψif and only ifL is invariant under the prolongation of the action.

Proof. For the first direction, assume that L satisfies (3.5). The Hamiltonian is H(q, p) =hF L(q, v), vi −L(q, v), where (q, p) =F L(q, v)

H(T^∗Ψ_g(q, p)) =H(T^∗Ψ_g◦F L(q, v))

=H(F L◦TΨ_g⁻¹(q, v))

=hF L◦TΨ_g−1(q, v), TqΨ_g−1vi −L(TΨ_g−1(q, v))

=hT^∗Ψg◦F L(q, v), TqΨg⁻¹vi −L(q, v)

=hF L(q, v), T_g·qΨg◦TqΨ_g−1vi −L(q, v)

=H(q, p).

For the converse statement, assume thatH(T^∗Ψ_g−1(q, p)) =H(q, p).

L(TΨ_g(q, v)) =L◦TΨ◦F H(q, p)

By F H = F L⁻¹, andT^∗Ψ_g−1 = (T^∗Ψ_g)⁻¹, and that the diagram commutes, it follows that this equals

L◦F H◦T^∗Ψ_g−1(q, p)

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and by (3.3) and (3.4)

L◦F H◦T^∗Ψ_g−1(q, p) =hT^∗Ψ_g−1(q, p), F H◦T^∗Ψ_g−1(q, p)i −H(T^∗Ψ_g−1(q, p))

= ∂

∂H((T^∗Ψ_g−1(q,(1 +)p))

₌₀−H(q, p)

= ∂

∂H(q,(1 +)p)

₌₀−H(q, p)

=hp, vi −H(q, p)

Where (q, v) =F H(q, p). The invariance ofLfollows.

3.4 Discrete Mechanics

The Lagrangian and Hamiltonian formulations of mechanics have corresponding formulations in a discrete setting. While the Hamiltonian H has no direct equivalent in a discrete setting, one can define a discrete LagrangianL_d :Q×Q→R. Specifically, in discrete Lagrangian mechanics one searches for a sequence of points (q1, q2, . . . , qn) which minimizes the discrete action

A(q1, . . . , q_n) =

n−1

X

i=1

L_d(q_i, q_i+1)

typically subject to constraints on the endpoints. Differentiating with respect to q_i leads to theDiscrete Euler–Lagrange equations

D2Ld(qi−1, qi) +D1Ld(qi, qi+1) = 0, (3.6) We define the two discrete Legendre transformsF^±Ld:Q×Q→T^∗Qby

F⁺Ld(qi, qi+1) =D2Ld(qi, qi+1) (3.7) F⁻Ld(qi, qi+1) =−D1Ld(qi, qi+1). (3.8) The discrete Euler–Lagrange equations can be stated in terms of the Legendre transforms as

F⁺L_d(q_i−1, q_i) =F⁻L_d(q_i, q_i+1) =p_i. Where the above equation defines the discrete momentump_i.

Under the assumption that the Legendre transforms are bijective, Ld is the generating function of the first kind of a symplectic map ˜F_L_d:T^∗Q→T^∗Q

F˜L_d=F⁺Ld◦(F⁻Ld)⁻¹ alternatively ˜FL_d: (qi, pi)7→(qi+1, pi+1) where

pi=−D1(qi, qi+1) p_i+1=D₂(q_i, q_i+1)

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3.4. DISCRETE MECHANICS 21 L_d also defines an advancement mapF_L_d:Q×Q→Q×Qby

FL_d= (F⁻Ld)⁻¹◦F⁺Ld. From the definitions of F^±Ld it follows that

F_L_d(q_i, q_i+1) = (q_i+1, q_i+2)

where (qi, qi+1, qi+2) satisfy the Discrete Euler–Lagrange equations (3.6).

Theorem 8. LetL_d:Q×Q→R, be invariant under the prolongation of the group action Ψ :G×Q→Q,

Ld(q0, q1) =Ld(Ψg(q0),Ψg(q1)). (3.9) The group action commutes with the discrete Legendre transforms in the sense that the following diagram commutes

Q×Q

Ψ^Q×Q_g

F^±Ld//T^∗Q

Q×Q

F^±Ld

//T^∗Q

T^∗Ψ_g

OO

Proof. For simpler notation, we write Ψ instead of Ψ^Q×Q. For the mapF⁺L_d, hT^∗Ψg◦F⁺Ld◦Ψg(qi, qi+1), vq_i+1i=hF⁺Ld(Ψg(qi),Ψg(qi+1)), Tq_i+1Ψgvq_i+1i

=hD2Ld(Ψg(qi),Ψg(qi+1)), Tqi+1Ψgvqi+1i

=hF⁺Ld(qi, qi+1), vqi+1i.

where the last line follows from applying the partial differentialD₂to the identity L=L◦Ψ^Q×Q_g and using that Ψ(q₁) does not depend onq₂. The proof for the map F⁻L_d is completely analogous.

The following theorem gives a sufficient condition for when maps generated from generating functions of the first kind are equivariant under a group action.

Theorem 9. If Ld :Q×Q→R is invariant under the prolongation of the group actionΨ :G×Q→Q, then the mapsFL_d:Q×Q→Q×QandF˜L_d:T^∗Q×T^∗Qare equivariant under the prolongation and the cotangent lift of the action, respectively.

Proof. The statement follows from combining the commuting diagrams forF⁺Ld

andF⁻Ld.

To relate the discrete and continuous formulations, one introduces the step- length parameter hin the discrete Lagrangian. The discretization of a continuous Lagrangian is a discrete Lagrangian

Ld(qi, qi+1, h)≈ Z h

0

L(q,q)dt,˙ (3.10)

Discrete Invariant Variational Problems

Master of Science in Physics and Mathematics

June 2011

Brynjulf Owren, MATH Submission date:

Supervisor:

Norwegian University of Science and Technology Department of Mathematical Sciences

Discrete Invariant Variational Problems

Geir Bogfjellmo

Problem Description

ABSTRACT

PREFACE

CONTENTS

CHAPTER 1 INTRODUCTION

CHAPTER 2

GROUP ACTIONS AND MOVING FRAMES

2.1 Basic Definitions and Results

2.2 Infinitesimals

2.3 Prolongation

2.4 Differentiation of Invariants

CHAPTER 3

MECHANICAL SYSTEMS

3.1 Symplectic Manifolds and Symplectic Maps

3.2 Lagrangian and Hamiltonian Systems

3.3 Symmetries

3.4 Discrete Mechanics