Relative Equilibria Of Hamiltonian Systems Research Articles

Stability transitions for axisymmetric relative equilibria of Euclidean symmetric Hamiltonian systems

In the presence of noncompact symmetry, the stability of relative equilibria under momentum-preserving perturbations does not generally imply robust stability under momentum-changing perturbations. For axisymmetric relative equilibria of Hamiltonian systems with Euclidean symmetry, we investigate different mechanisms of stability: stability by energy–momentum confinement, KAM, and Nekhoroshev stability, and we explain the transitions between them. We apply our results to the Kirchhoff model for the motion of an axisymmetric underwater vehicle, and we numerically study dissipation induced instability of KAM stable relative equilibria for this system.

Stability of relative equilibria with singular momentum values in simple mechanical systems

A method for testing the stability of relative equilibria in Hamiltonian systems of the form ‘kinetic + potential energy’ is presented. This method extends previously existing techniques to the case of non-free group actions and singular momentum values. A normal of the symplectic matrix at a relative equilibrium is also obtained.

Stability of Poisson Equilibria and Hamiltonian Relative Equilibria by Energy Methods

We develop a general stability theory for equilibrium points of Poisson dynamical systems and relative equilibria of Hamiltonian systems with symmetries, including several generalisations of the Energy-Casimir and Energy-Momentum Methods. Using a topological generalisation of Lyapunov’s result that an extremal critical point of a conserved quantity is stable, we show that a Poisson equilibrium is stable if it is an isolated point in the intersection of a level set of a conserved function with a subset of the phase space that is related to the topology of the symplectic leaf space at that point. This criterion is applied to generalise the energy-momentum method to Hamiltonian systems which are invariant under non-compact symmetry groups for which the coadjoint orbit space is not Hausdorff. We also show that a G-stable relative equilibrium satisfies the stronger condition of being A-stable, where A is a specific group-theoretically defined subset of G which contains the momentum isotropy subgroup of the relative equilibrium. The results are illustrated by an application to the stability of a rigid body in an ideal irrotational fluid.

Persistence of relative equilibria in Hamiltonian systems with non-compact symmetry

We prove new results on the persistence of Hamiltonian relative equilibria with generic velocity–momentum pairs in the case of non-compact non-free group actions and taking into account time reversibility. Our starting point is a relative equilibrium which is non-degenerate modulo isotropy which, in the case of a generic momentum implies persistence of the given relative equilibrium to all nearby momentum values with the same isotropy. We show that the analysis of the persistence problem involves the study of a singular algebraic variety which is determined solely by the symmetry group of the problem. We present persistence results for relative equilibria with velocity–momentum pairs which are regular points of this variety and give sufficient conditions for a velocity–momentum pair to be regular. We apply our results to relative equilibria of Euclidean equivariant systems, including models of rigid bodies in fluids.

Hamiltonian Systems near Relative Equilibria

We give explicit differential equations for the dynamics of Hamiltonian systems near relative equilibria. These split the dynamics into motion along the group orbit and motion inside a slice transversal to the group orbit. The form of the differential equations that is inherited from the symplectic structure and symmetry properties of the Hamiltonian system is analysed and the effects of time reversing symmetries are included. The results will be applicable to the stability and bifurcation theories of relative equilibria of Hamiltonian systems.

Unstable manifolds of relative equilibria in Hamiltonian systems with dissipation

This paper studies the destabilizing effects of dissipation on familiesof relative equilibria in Hamiltonian systems which are non-extremalconstraint critical points in the energy-Casimir or theenergy-momentum methods. The dissipation is allowed to destroy theconservation law associated with the symmetry group or Casimirs, as longas the family of relative equilibria stays on an invariant manifold. Thisapproach complements earlier work in the literature, in which thedissipation did not affect the conservation law. Firstly, Chetaev's instability theorem is extended to invariantmanifolds and this extended theorem is used to prove the instability offamilies of relative equilibria for several examples. Secondly, it is shownthat families of non-extremal stationary solutions of the two-dimensionalincompressible homogeneous Euler equations are unstable for thecorresponding viscous perturbations of this system, i.e. for thetwo-dimensional Navier-Stokes equations. Also, the instability ofthe sleeping top relative equilibria under friction can easily be provedin this way, even before the Hamiltonian sleeping top becomes linearlyunstable. Finally, sufficient conditions are given for which friction destabilizes families of non-extremal relative equilibria in simple mechanical systems with Abelian symmetry.

Stability of Hamiltonian relative equilibria

We generalize a sufficient condition for the stability of relative equilibria in symmetric Hamiltonian systems, due to Patrick (1992 Relative equilibria in Hamiltonian systems: the dynamic interpretation of nonlinear stability on a reduced phase space J. Geo. Phys. 9 111-19), to the case in which these relative equilibria have non-trivial symmetry. We also describe a block diagonalization that facilitates the use of this result in particular examples and shows the relation between the stability of the relative equilibrium and the Lyapunov stability of the associated singular reduced equilibrium.

Bifurcations from relative equilibria of Hamiltonian systems

A symplectic version of the slice theorem for compact group actions is used to give a general description of the dynamics of a symmetric Hamiltonian system near a relative equilibrium as an interaction between rotational and vibrational motion. This is then used to prove a bifurcation theorem for relative equilibria. The bifurcation theorem is applied to two examples, the classical dynamics of an molecule near a symmetric linear equilibrium, and the dynamics of a system of two coupled identical axisymmetric rigid bodies near an equilibrium for which the two bodies are aligned on top of each other. Slice reduction of these systems is also used to describe other aspects of their dynamics. The results suggest that this will be a useful general technique for describing the dynamics of systems near relative equilibria which are singular points of the associated momentum maps.

Approximations with curves of relative equilibria in Hamiltonian systems with dissipation

In this paper we will investigate the relevance of a stable family of relative equilibria in a dissipative Hamiltonian system with symmetry. We are interested in relative equilibria of the Hamiltonian system, whose stability follows from the fact that they are local extrema of the energy-momentum function which is a combination of the Hamiltonian and a conserved quantity of the Hamiltonian system, induced by the momentum map related to the symmetry group. Although the dissipative perturbation is equivariant under the action of the symmetry group, it will destroy the conservation law associated with the symmetry group. We will specify its dissipative properties in terms of the induced time behaviour of the momentum map and quasi-static attractive properties of the relative equilibria. By analysing the time behaviour of the previously mentioned energy-momentum function we derive sufficient conditions such that solutions of the dissipative system which are initially close to a relative equilibrium can be approximated by a (long) curve of relative equilibria. At the end we illustrate the method by analysing the example of a rigid body in a rotational symmetric field with dissipative rotation-like perturbation added.

Relative equilibria of Hamiltonian systems with symmetry: Linearization, smoothness, and drift

A relative equilibrium of a Hamiltonian system with symmetry is a point of phase space giving an evolution which is a one-parameter orbit of the action of the symmetry group of the system. The evolutions of sufficiently small perturbations of a formally stable relative equilibrium are arbitrarily confined to that relative equilibrium's orbit under the isotropy subgroup of its momentum. However, interesting evolution along that orbit, here called drift, does occur. In this article, linearizations of relative equilibria are used to construct a first order perturbation theory explaining drift, and also to determine when the set of relative equilibria near a given relative equilibrium is a smooth symplectic submanifold of phase space.

Relative equilibria in Hamiltonian systems: The dynamic interpretation of nonlinear stability on a reduced phase space

Stability of relative equilibria for Hamiltonian systems is generally equated with Liapunov stability of the corresponding fixed point of the flow on the reduced phase space. Under mild assumptions, a sharp interpretation of this stability is given in terms of concepts on the unreduced space.