Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 038, 28 pages      arXiv:1210.0278      https://doi.org/10.3842/SIGMA.2013.038
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”

Relative Critical Points

Debra Lewis
Mathematics Department, University of California, Santa Cruz, Santa Cruz, CA 95064, USA

Received October 01, 2012, in final form May 06, 2013; Published online May 17, 2013

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.

Key words: relative equilibria; symmetries; conservative systems; Riemann ellipsoids.

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