Relative equilibria in systems with configuration space isotropy

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Abstract

We present a framework for studying Lagrangian and Hamiltonian systems with symmetries, near points with configuration space isotropy. We use twisted parametrisations corresponding to phase space slices based at zero points of (co-)tangent fibres. Given a hyperregular Lagrangian, we find a Legendre transform in the twisted coordinates. For simple mechanical systems, we state necessary and sufficient conditions for the existence of relative equilibria in terms of an augmented-amended potential.

Introduction

The study of symmetry in mechanical systems has a very long history, going back to the foundations of classical mechanics. Typical questions include: how can we exploit the symmetry to simplify the problem? To what degree can we separate rotational from vibrational motion? What are the simplest or generic dynamical behaviors of a symmetrical system? The simplest non-equilibrium solution of a symmetrical system is a relative equilibrium, which is a solution that moves only in a symmetry direction; an example is the steady spinning of a rigid body around one of its principal axes. Like equilibria in general, these solutions can be used as organizing centres for understanding more complex dynamics.

In recent decades, symmetry has received particular attention in the field of geometric mechanics. Some key achievements are: the theory of symplectic reduction (see for example [1], [2], [19], [33], [26]); the Marle–Guillemin–Sternberg normal form (the Hamiltonian slice theorem) [12], [6]; and the energy-momentum method [27], [13], [10], [22]. These and many related results have given a firm geometrical foundation to the study of symmetry of Hamiltonian systems on symplectic manifolds. Many important mechanical systems have phase space which are (co-)tangent bundles, with (co-)tangent lifted symmetries. The geometrical theory of symmetry specific to (co-)tangent bundles has also seen many recent advances (see for example [11], [15], [13], [16], [5], [32], [29]). However, our understanding of lifted symmetries with non-trivial isotropy is far from complete.

The present paper considers Lagrangian and Hamiltonian systems on (co-)tangent bundles, with lifted symmetries and configuration-space isotropy. We present a practical geometrical framework for studying such systems using degenerate parametrisations of neighbourhoods of phase space points with configuration-space isotropy. The parametrisations are defined in Sections 2 The Lagrangian side, 3 The Hamiltonian side; they are tubes (defined below) around zero points in the (co-)tangent fibres. We find a Legendre transform in the new coordinates (Section 4), which allows us to study relative equilibria of Lagrangian systems in a Hamiltonian context. Finally, we study simple mechanical systems (Section 6), giving a set of necessary and sufficient conditions for the existence of a relative equilibrium, expressed in terms of an augmented-amended potentialwhich generalises both the augmented potential and amended potential familiar from systems with free symmetries (see [13]).

For the motivation of this paper we are indebted to a series of molecular physicists and chemists, going back at least to Watson [34]. Our results in Sections 2 The Lagrangian side, 3 The Hamiltonian side, 4 Legendre transforms provide a theoretical foundation for techniques they have used in particular examples. Our geometrical formulation of these techniques builds on the presentation in [9].

We begin by summarising relevant basic facts about Lie group symmetries, including Palais’ slice theorem (see [1], [3], [7], [4], [14], [26]). The starting point of our approach will be to apply the slice theorem in a configuration space Q. We will then use the resulting parametrisation to study the phase spaces TQand TQ.

Let G be a Lie group, with Lie algebra g, and consider a smooth left action of G on a finite-dimensional manifold M, written (g,q)gq. For every ξg and zM, the infinitesimal action of ξ on z isξz=ddtexp(tξ)zt=0.The isotropy subgroup of zM is Gz:={gG|gz=z}. An action is free if all of the isotropy subgroups Gz are trivial.

An action is proper if the map g,z(z,gz) is proper (i.e. the preimage of every compact set is compact). Note that this is always the case if G is compact. A key elementary property of proper actions is that all isotropy subgroups are compact. If G acts properly and freely on M, then M/G has a unique smooth structure such that πG:MM/G is a submersion (in fact, πG is a principal bundle). One useful consequence is that for every zM, we have kerTzπG=Tz(Gz)=gz.

Given a G action Φ:G×QQ, the group G has a tangent lift action on TQ, given by gv=TΦg(v), and a cotangent lift action on TQ, given by gα=(TΦg1)α. In this context, the space Q called the configuration space or base space. The tangent or cotangent lift of a proper (resp. free) action is proper (resp. free). For any qQ, the isotropy group Gq is called the configuration space isotropy of any point vTqQ or zTqQ. The cotangent bundle TQ has a canonical symplectic form, given in given local coordinates by ω=dqidpi. Every cotangent-lifted action on TQ is symplectic with respect to this symplectic form and has an Ad-equivariant momentum map given by J(αq),ξ=αq,ξq.

Let K be a Lie subgroup of G, and S is a manifold on which K acts. Consider the following two left actions on G×S:Kacts bytwisting:k(g,s)=(gk1,ks)Gacts by leftmultiplication:γ(g,s)=(γg,s).It is easy to show that these actions are free and proper and commute. The twisted product G×KS is the quotient of G×S by the twist action of K. It is a smooth manifold; in fact G×KSG/K is the vector bundle associated to the K action on S. The left multiplication action of G commutes with the twist action and drops to a smooth G action on G×KS, given by γ[g,s]K=[γg,s]K.

Now consider a G action on a manifold M, and a point zM, and let K=Gz be the isotropy subgroup of z. A tube for the G action at z is a G-equivariant diffeomorphism from some twisted product G×KS to an open neighbourhood of Gz in M, that maps [e,0]K to z. The space S may be embedded in G×KS as {[e,s]K:sS}; the image of the latter by the tube is called a slice.

The slice theorem of Palais [28] states that tubes always exist for smooth proper actions of a Lie group G on manifold M. One version of the theorem is as follows. Given zM, with isotropy group K=Gz, there always exists a G-invariant Riemannian metric on a neighbourhood of z. Let N be the orthogonal complement gz. Then there exists a K-invariant neighbourhood S of 0 in N such that the mapτ:G×KSM[g,s]Kgexpzs(where expz is the Riemannian exponential) is a tube for the G action at z. The K-invariant complement N to gz is sometimes called a l inear slice to the G action at z. The twisted product G×KN may be identified with the normal bundle to the orbit Gz. If the G action is linear, then we can replace “expzs” with “(z+s)” in the above statement, and S may be chosen to be any neighbourhood of 0 such that τ is injective.

Consider a Lagrangian L:TQR, invariant under a proper tangent-lifted action of a Lie group G. Let q0Q and K=Gq0.

We can apply Palais’s slice theorem around q0, giving a tube

τ:G×KSQ[g,s]Kgexpq0sIf Q is an open subset of a vector space, with G acting linearly, as, for example, in gravitational and molecular N-body problems, then S can be identified with a neighbourhood of the origin in a linear subspace of Q itself, and the tube defined by τ([g,s]K)=g(q0+s). In any case, pulling back τ by the projection πK:G×SG×KS gives a map τπK:G×SQ which we regard as degenerate “parametrisation” of Q in a neighbourhood of q, defining the “slice coordinates” (g,s). This parametrisation is semi-global in the sense that it is global in the group direction and local in the slice direction. The tangent and cotangent lifts of πKτ give parametrisations T(G×S)TQ and T(G×S)TQ. In this paper we will describe mechanical systems on TQ and TQ, with configurations in the neighbourhood of the group orbit Gq0, by pulling them back to T(G×S) and T(G×S). We now describe the actions of G and K on these spaces.

Let g and k be the Lie algebras of G and K. Throughout the paper we identify TG with G×g and TG with G×g using the trivialisations given by:TGG×gandTGG×gTLgζ(g,ζ)TLg1μ(g,μ)where Lg is left multiplication by g. Similarly, we make the identificationsT(G×S)TG×TSG×g×TST(G×S)TG×TSG×g×TS

Note that TS and TS are trivial, as S is a subset of a vector space. We write elements of TS as (s,s˙) and elements of TS as (s,σ). The left multiplication action of G and twist action of K on G×S lift to free, proper, commuting actions on T(G×S) and T(G×S). In the above trivialisations, the lifted actions are:tangent lifted twist:k(g,ζ,s,s˙)=(gk1,Adkζ,ks,ks˙)cotangent lifted twist:k(g,μ,s,σ)=(gk1,Adk1μ,ks,kσ)tangent lifted left multiplication:γ(g,ζ,s,s˙)=(γg,ζ,s,s˙)cotangent lifted left multiplication:γ(g,μ,s,σ)=(γg,μ,s,σ).The corresponding infinitesimal actions of ξk and ηg are:tangent lifted twist:ξ(g,ζ,s,s˙)=(ξ,adξζ,ξs,ξs˙)cotangent lifted twist:ξ(g,μ,s,σ)=(ξ,Adξμ,ξs,ξσ)tangent lifted left multiplication:η(g,ζ,s,s˙)=(Adg1η,0,0,0)cotangent lifted left multiplication:η(g,μ,s,σ)=(Adg1η,0,0,0).

The cotangent-lifted actions have the following momentum maps, with respect to the canonical symplectic form on T(G×S):twist:JK(g,μ,s,σ)=μ|k+JS(s,σ)=μ|k+sσkleft multiplication:JG(g,μ,s,σ)=Adg1μg,where sσ is defined by sσ,ξ=σ,ξs for all ξk and is the momentum map JS:TSk for the action of K on TS. The momentum map JK is equivariant with respect to the twist action of K and invariant under the left multiplication action of G, while JG is equivariant with respect to the action of G and invariant under the twist action.

For simplicity of notation, we will sometimes identify Q with G×KS and q0 with [e,0]K.

Section snippets

The Lagrangian side

The goal of this section is to describe the tangent bundle T(G×KS) and any Lagrangian system on it, using a parametrisation by G×k×TST(G×S).

Fix a K-invariant complement of k in g, which we denote k (such a complement can always be found by averaging over K, since K is compact). Consider the projection πK:G×SG×KS. Its tangent map is a K-invariant G-equivariant surjection. If we describe points in T(G×KS) as TπK(g,ζ,s,s˙), then we have two kinds of degeneracy in our coordinates: first, (g,s)

The Hamiltonian side

In this section we describe a parametrisation of T(G×KS) by G×k×TS that is dual to the parametrisation of T(G×KS) by G×k×TS described in the previous section. Here k is the annihilator of k in g. Note that our choice of splitting g=kk induces a dual splitting g=kkk(k).

Recall that JK is the momentum map for the cotangent-lifted twist action of K on T(G×S). By regular cotangent bundle reduction, the symplectic reduced space at zero JK1(0)/K is symplectomorphic to T((G×S)/K)=T(G

Legendre transforms

In the previous two sections we have found twisted parametrisations of TQ and TQ respecting a given symmetry. Given an original symmetric Lagrangian on TQ or symmetric Hamiltonian on TQ, we have defined “lifted Lagrangians” and “lifted Hamiltonians” using the new parametrisations. In the present section we consider the situation when the original Hamitonian is obtained from the original Lagrangian as the Legendre transform of the energy. We obtain Legendre transforms of the lifted

Hamilton’s equations and relative equilibria

In this section we outline a calculation of Hamilton’s equations in the twisted parametrisation of TQ given in Section 3. The result will be a special case of the bundle equationsor r econstruction equations (see Remark 4). We then apply these to give conditions for the existence of relative equilibria.

Recall that, given any K-invariant extension Hext of the function H˜Hρ, the restriction of the Hamiltonian vector field XHext to JK1(0) projects down to the orignal Hamiltonian vector field XH

Simple mechanical systems

In this section we consider the special case of a simple mechanical system, which is one in which the Lagrangian L:TQR has the formL(q,vq)=12K(vq,vq)V(q)for some G-invariant Riemannian metric K on TQ, called the kinetic energy, and some G-invariant potential V:QR. We compute the Lagrangian L¯ and Hamiltonian H¯ for such systems and use H¯ together with the results of the previous section to give conditions for the existence of relative equilibria of simple mechanical systems.

Let q0Q have

Comments

We have outlined a framework for studying mechanical systems determined by a symmetric Lagrangian on TQ or a symmetric Hamiltonian on TQ, at configurations near a given one, q0, with nontrivial isotropy. We have found tubes around “(q0,0)”, meaning 0Tq0Q or 0Tq0Q, which we consider as twisted parametrisations for TQ and TQ. These parametrisations are semi-local in the configuration space but global in the fibre direction. This means that we can study all local dynamics near a given

Acknowledgement

This research was supported by European Community funding for the Research Training MASIE (HPRN-CT-2000-0013). T.S. thanks the University of Surrey for their support.

References (27)

  • E. Lerman et al.

    Stability and persistence of relative equilibria at singular values of the moment map

    Nonlinearity

    (1998)
  • D. Lewis

    Lagrangian block diagonalization

    J. Dyn. Differential Equations

    (1992)
  • C.-M. Marle, Modèle d’action hamiltonienne d’un groupe de Lie sur une variété symplectique, vol. 43 (2), Rendiconti del...
  • Cited by (12)

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