The EulerPoincare Equations and Semidirect Products with Applications to Continuum Theories
Abstract
We study EulerPoincare systems (i.e., the Lagrangian analogue of LiePoisson Hamiltonian systems) defined on semidirect product Lie algebras. We first give a derivation of the EulerPoincare equations for a parameter dependent Lagrangian by using a variational principle of Lagrange d'Alembert type. Then we derive an abstract KelvinNoether theorem for these equations. We also explore their relation with the theory of LiePoisson Hamiltonian systems defined on the dual of a semidirect product Lie algebra. The Legendre transformation in such cases is often not invertible; so it does not produce a corresponding EulerPoincare system on that Lie algebra. We avoid this potential difficulty by developing the theory of EulerPoincare systems entirely within the Lagrangian framework. We apply the general theory to a number of known examples, including the heavy top, ideal compressible fluids and MHD. We also use this framework to derive higher dimensional CamassaHolm equations, which have many potentially interesting analytical properties. These equations are EulerPoincare equations for geodesics on diffeomorphism groups (in the sense of the Arnold program) but where the metric is H^1 rather than L^2.
 Publication:

arXiv eprints
 Pub Date:
 January 1998
 DOI:
 10.48550/arXiv.chaodyn/9801015
 arXiv:
 arXiv:chaodyn/9801015
 Bibcode:
 1998chao.dyn..1015H
 Keywords:

 Chaotic Dynamics;
 Nonlinear Sciences  Chaotic Dynamics
 EPrint:
 72 pages, LATeX, no figures