Relative Critical Points

TL;DR

This paper explores the concept of relative equilibria in symmetric dynamical systems, proposing a unified approach using invariant functions to analyze critical points across various mechanical systems.

Contribution

It introduces a method to construct fully invariant functions on the product manifold, enabling analysis of critical points without relying on parametrized families.

Findings

Unified framework for analyzing relative equilibria

Application to classical mechanical systems like the Lagrange top and rigid body

Extension of methods to non-Hamiltonian settings

Abstract

Relative equilibria of Lagrangian and Hamiltonian systems with symmetry are critical points of appropriate scalar functions parametrized by the Lie algebra (or its dual) of the symmetry group. Setting aside the structures - symplectic, Poisson, or variational - generating dynamical systems from such functions highlights the common features of their construction and analysis, and supports the construction of analogous functions in non-Hamiltonian settings. If the symmetry group is nonabelian, the functions are invariant only with respect to the isotropy subgroup of the given parameter value. Replacing the parametrized family of functions with a single function on the product manifold and extending the action using the (co)adjoint action on the algebra or its dual yields a fully invariant function. An invariant map can be used to reverse the usual perspective: rather than selecting a parametrized family of functions and finding their critical points, conditions under which functions will be critical on specific orbits, typically distinguished by isotropy class, can be derived. This strategy is illustrated using several well-known mechanical systems - the Lagrange top, the double spherical pendulum, the free rigid body, and the Riemann ellipsoids - and generalizations of these systems.