On the Stability of Relative Equilibria

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Abstract
This work concerns the variational characterization of stability properties of relative equilibria. In particular, tests bearing upon the second variation in constrained variational principles are applied to prove stability of certain symmetry-related solutions of noncanonical Hamiltonian systems. The new approach is essentially an appropriate extension of the principle of exchange of stability to conditional variational principles in which the Lagrange multipliers are viewed as bifurcation parameters. The technique is introduced in the context of the classic problem of a heavy asymmetric rigid-body moving about a fixed point. In this example, the special solutions are known as the axes of Staude. The example is also used to elucidate issues concerning the necessity of stability conditions obtained by variational means. Finally, a discussion is presented of the behaviour of variational stability estimates as the symmetry class of the problem is altered.

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