George W. Patrick

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

Abstract 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.

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

SummaryA 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 equilibriums 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.

Error analysis of variational integrators of unconstrained Lagrangian systems

An error analysis of variational integrators is obtained, by blowing up the discrete variational principles, all of which have a singularity at zero time-step. Divisions by the time step lead to an order that is one less than observed in simulations, a deficit that is repaired with the help of a new past–future symmetry.

A Symplectic Integrator for Riemannian Manifolds

SummaryThe configuration spaces of mechanical systems usually support Riemannian metrics which have explicitly solvable geodesic flows and parallel transport operators. While not of primary interest, such metrics can be used to generate integration algorithms by using the known parallel transport to evolve points in velocity phase space.

The transversal relative equilibria of a Hamiltonian system with symmetry

Let P be a symplectic manifold with a free symplectic action of a connected compact Lie group G. We show that, given a certain transversality condition, the set of relative equilibria near pe of a G-invariant Hamiltonian system on P is locally Whitney-stratified by the conjugacy classes of the isotropy subgroups (under the product of the coadjoint and adjoint actions) of the momentum-generator pairs (µ,ξ) of the relative equilibria. The dimension of the stratum of the conjugacy class (K) is dim G + 2dim Z(K)-dim K, where Z(K) is the centre of K. Transverse to this stratum is locally diffeomorphic to the set of commuting pairs of the Lie algebra of K/Z(K). The stratum (K) is a symplectic submanifold of P near pe if and only if pe is non-degenerate and K is a maximal torus of G. We also show that the set of G-invariant Hamiltonians on P for which all the relative equilibria are transversal is open and dense. Thus, generically, the types of singularities of the set of relative equilibria of a Hamiltonian system with symmetry are those types found amongst the singularities at zero of the sets of commuting pairs of certain Lie subalgebras of the symmetry group.

Dynamics of Perturbed Relative Equilibria of Point Vortices on the Sphere or Plane

SO(3) , and there are stable relative equilibria of four point vortices, where three identical point vortices form an equilateral triangle circling a central vortex. These relative equilibria have zero (nongeneric) momentum and form a family that extends to arbitrarily small diameters. Using the energy-momentum method, I show their shape is stable while their location on the sphere is unstable, and they move, after perturbation to nonzero momentum, on the sphere as point particles move under the influence of a magnetic monopole. In the analysis the internal and external degrees of freedom are separated and the mass of these point particles determined. In addition, two identical such relative equilibria attract one another, while opposites repel, and in energetic collisions, opposites disintegrate to vortex pairs while identicals interact by exchanging a vortex. An analogous situation also occurs for the planar system with its noncompact SE(2) symmetry.

Dynamics near relative equilibria: Nongeneric momenta at a 1:1 group-reduced resonance

Abstract. An interesting situation occurs when the linearized dynamics of the shape of a formally stable Hamiltonian relative equilibrium at nongeneric momentum 1:1 resonates with a frequency of the relative equilibriums generator. In this case some of the shape variables couple to the group variables to first order in the momentum perturbation, and the first order perturbation theory implies that the relative equilibrium slowly changes orientation in the same way that a charged particle with magnetic moment moves on a sphere under the influence of a radial magnetic monopole. In the course of showing this a normal form is constructed for linearizations of relative equilibria and for Hamiltonians near group orbits of relative equilibria.

Stability by KAM Confinement of Certain Wild, Nongeneric Relative Equilibria of Underwater Vehicles with Coincident Centers of Mass and Buoyancy ∗

Purely rotational relative equilibria of an ellipsoidal underwater vehicle occur at nongeneric momentum where the symplectic reduced spaces change dimension. The stability of these relative equilibria under momentum changing perturbations is not accessible by Lyapunov functions obtained from energy and momentum. A blow-up construction transforms the stability problem to the analysis of symmetry-breaking perturbations of Hamiltonian relative equilibria. As such, the stability follows by KAM theory rather than energy-momentum confinement.

Skew critical problems

Skew critical problems occur in continuous and discrete nonholonomic Lagrangian systems. They are analogues of constrained optimization problems, where the objective is differentiated in directions given by an apriori distribution, instead of tangent directions to the constraint. We show semiglobal existence and uniqueness for nondegenerate skew critical problems, and show that the solutions of two skew critical problems have the same contact as the problems themselves. Also, we develop some infrastructure that is necessary to compute with contact order geometrically, directly on manifolds.

Geometric discrete analogues of tangent bundles and constrained Lagrangian systems

Abstract Discretizing variational principles, as opposed to discretizing differential equations, leads to discrete-time analogues of mechanics, and, systematically, to geometric numerical integrators. The phase space of such variational discretizations is often the set of configuration pairs, analogously corresponding to initial and terminal points of a tangent vector. We develop alternative discrete analogues of tangent bundles, by extending tangent vectors to finite curve segments, one curve segment for each tangent vector. Towards flexible, high order numerical integrators, we use these discrete tangent bundles as phase spaces for discretizations of the variational principles of Lagrangian systems, up to the generality of nonholonomic mechanical systems with nonlinear constraints. We obtain a self-contained and transparent development, where regularity, equations of motion, symmetry and momentum, and structure preservation, all have natural expressions.