Relative equilibria in systems with configuration space isotropy
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 .
Let G be a Lie group, with Lie algebra , and consider a smooth left action of G on a finite-dimensional manifold M, written . For every and , the infinitesimal action of on z isThe isotropy subgroup of is . An action is free if all of the isotropy subgroups are trivial.
An action is proper if the map 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 has a unique smooth structure such that is a submersion (in fact, is a principal bundle). One useful consequence is that for every , we have .
Given a G action , the group G has a tangent lift action on TQ, given by , and a cotangent lift action on , given by . 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 , the isotropy group is called the configuration space isotropy of any point or . The cotangent bundle has a canonical symplectic form, given in given local coordinates by . Every cotangent-lifted action on is symplectic with respect to this symplectic form and has an -equivariant momentum map given by .
Let K be a Lie subgroup of G, and S is a manifold on which K acts. Consider the following two left actions on :It is easy to show that these actions are free and proper and commute. The twisted product is the quotient of by the twist action of K. It is a smooth manifold; in fact 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 , given by .
Now consider a G action on a manifold M, and a point , and let be the isotropy subgroup of z. A tube for the G action at z is a G-equivariant diffeomorphism from some twisted product to an open neighbourhood of in M, that maps to z. The space S may be embedded in as ; 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 , with isotropy group , there always exists a G-invariant Riemannian metric on a neighbourhood of z. Let N be the orthogonal complement . Then there exists a K-invariant neighbourhood S of 0 in N such that the map(where is the Riemannian exponential) is a tube for the G action at z. The K-invariant complement N to is sometimes called a l inear slice to the G action at z. The twisted product may be identified with the normal bundle to the orbit . If the G action is linear, then we can replace “” with “()” in the above statement, and S may be chosen to be any neighbourhood of 0 such that is injective.
Consider a Lagrangian , invariant under a proper tangent-lifted action of a Lie group G. Let and .
We can apply Palais’s slice theorem around , giving a tube
If 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 . In any case, pulling back by the projection gives a map which we regard as degenerate “parametrisation” of Q in a neighbourhood of q, defining the “slice coordinates” . 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 give parametrisations and . In this paper we will describe mechanical systems on and , with configurations in the neighbourhood of the group orbit , by pulling them back to and . We now describe the actions of G and K on these spaces.
Let and be the Lie algebras of G and K. Throughout the paper we identify TG with and with using the trivialisations given by:where is left multiplication by g. Similarly, we make the identifications
Note that TS and are trivial, as S is a subset of a vector space. We write elements of TS as and elements of as . The left multiplication action of G and twist action of K on lift to free, proper, commuting actions on and . In the above trivialisations, the lifted actions are:The corresponding infinitesimal actions of and are:
The cotangent-lifted actions have the following momentum maps, with respect to the canonical symplectic form on :where is defined by for all and is the momentum map for the action of K on . The momentum map is equivariant with respect to the twist action of K and invariant under the left multiplication action of while 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 and with .
Section snippets
The Lagrangian side
The goal of this section is to describe the tangent bundle and any Lagrangian system on it, using a parametrisation by .
Fix a K-invariant complement of in , which we denote (such a complement can always be found by averaging over K, since K is compact). Consider the projection . Its tangent map is a K-invariant G-equivariant surjection. If we describe points in as , then we have two kinds of degeneracy in our coordinates: first,
The Hamiltonian side
In this section we describe a parametrisation of by that is dual to the parametrisation of by described in the previous section. Here is the annihilator of in . Note that our choice of splitting induces a dual splitting .
Recall that is the momentum map for the cotangent-lifted twist action of K on . By regular cotangent bundle reduction, the symplectic reduced space at zero is symplectomorphic to
Legendre transforms
In the previous two sections we have found twisted parametrisations of TQ and respecting a given symmetry. Given an original symmetric Lagrangian on TQ or symmetric Hamiltonian on , 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 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 of the function , the restriction of the Hamiltonian vector field to projects down to the orignal Hamiltonian vector field
Simple mechanical systems
In this section we consider the special case of a simple mechanical system, which is one in which the Lagrangian has the formfor some G-invariant Riemannian metric on TQ, called the kinetic energy, and some G-invariant potential . We compute the Lagrangian and Hamiltonian for such systems and use together with the results of the previous section to give conditions for the existence of relative equilibria of simple mechanical systems.
Let have
Comments
We have outlined a framework for studying mechanical systems determined by a symmetric Lagrangian on or a symmetric Hamiltonian on , at configurations near a given one, , with nontrivial isotropy. We have found tubes around “”, meaning or , which we consider as twisted parametrisations for and . 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.
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