Nonlinear Mechanics and Applied Analysis
Abstract
Many important Hamiltonian systems have periodic solutions that are associated with symmetries of the equations. While it is well known that stationary solutions of a Hamiltonian system can be characterized as extremals of the potential energy, it is less widely appreciated that symmetry-related periodic solutions, or relative equilibria, can also be given a variational characterization, typically involving constraints. This variational characterization is important because if a periodic solution is associated with a constrained minimizer (in some sense), as opposed to merely being a stationary point, then a stability result is very often available. We are therefore left with the problem of characterizing those extremals of a constrained variational principle that are actually constrained local minima. It is shown how to apply the new results in the special context of Hamiltonian mechanics, and various stability and instability theorems are described. The machinery developed here can be viewed as an alternative to the energy-casimir and energy-momentum methods with the benefit that the necessary tests can be concretely and rigorously applied in several complex examples of physical importance.
Document Details
- Document Type
- Technical Report
- Publication Date
- Oct 31, 1991
- Accession Number
- ADA243978
Entities
People
- John H. Maddocks
Organizations
- University of Maryland