We study Euler–Poincaré systems (i.e., the Lagrangian analogue of Lie-Poisson Hamiltonian systems) defined on semidirect product Lie algebras. We first give a derivation of the Euler–Poincaré equations for a parameter dependent Lagrangian by using a variational principle of Lagrange d’Alembert type. Then we derive an abstract Kelvin-Noether theorem for these equations. We also explore their relation with the theory of Lie-Poisson Hamiltonian systems defined on the dual of a semidirect product Lie algebra. The Legendre transformation in such cases is often not invertible; thus, it does not produce a corresponding Euler–Poincaré system on that Lie algebra. We avoid this potential difficulty by developing the theory of Euler–Poincaré 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 Camassa-Holm equations, which have many potentially interesting analytical properties. These

Low’s well known action principle for the Maxwell-Vlasov equations of ideal plasma dynamics was originally expressed in terms of a mixture of Eulerian and Lagrangian variables. By imposing suitable constraints on the variations and analyzing invariance properties of the Lagrangian, as one does for the Euler equations for the rigid body and ideal fluids, we first transform this action principle into purely Eulerian variables. Hamilton’s principle for the Eulerian description of Low’s action principle then casts the Maxwell-Vlasov equations into Euler-Poincaré form for right invariant motion on the diffeomorphism group of position-velocity phase space, R 6. Legendre transforming the Eulerian form of Low’s action principle produces the Hamiltonian formulation of these equations in the Eulerian description. Since it arises from Euler-Poincaré equations, this Hamiltonian formulation can be written in terms of a Poisson structure that contains the Lie-Poisson bracket on the dual of a semidirect product Lie algebra. Because of

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–Poincaré 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-Poincaré form and discuss their corresponding Kelvin–Noether theorems and potential vorticity conservation laws. The various levels of GFD approximation are related

There is a well developed and useful theory of Hamiltonian reduction for semidirect products, which applies to examples such as the heavy top, compressible fluids and MHD, which are governed by Lie-Poisson type equations. In this paper we study the Lagrangian analogue of this process and link it with the general theory of Lagrangian reduction; that is the reduction of variational principles. These reduced variational principles are interesting in their own right since they involve constraints on the allowed variations, analogous to Research partially supported by NSF grant DMS 96–33161 and DOE contract DE–FG0395–

This paper was also one of the first to notice deep links between reduction and integrable systems, a subject continued by, for example, Bobenko, Reyman and Semenov-Tian-Shansky [1989].

Extreme events in solutions of hydrostatic and non-hydrostatic climate models

Extreme events in solutions of hydrostatic and non-hydrostatic climate models

1.1 Road map for the course................................... 7