Nonlinear Sciences – Chaotic Dynamics
Scientific paper
1999-03-25
Nonlinear Sciences
Chaotic Dynamics
42 pages, no figures, Isaac Newton Institute Proceedings (to appear)
Scientific paper
Recent theoretical work has developed the Hamilton's-principle analog of Lie-Poisson Hamiltonian systems defined on semidirect products. The main theoretical results are twofold: (1) Euler-Poincar\'e equations (the Lagrangian analog of Lie-Poisson Hamiltonian equations) are derived for a parameter dependent Lagrangian from a general variational principle of Lagrange d'Alembert type in which variations are constrained; (2) an abstract Kelvin-Noether theorem is derived for such systems. By imposing suitable constraints on the variations and by using invariance properties of the Lagrangian, as one does for the Euler equations for the rigid body and ideal fluids, we cast several standard Eulerian models of geophysical fluid dynamics (GFD) at various levels of approximation into Euler-Poincar\'{e} form and discuss their corresponding Kelvin-Noether theorems and potential vorticity conservation laws.
Holm Darryl D.
Marsden Jerrold E.
Ratiu Tudor S.
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