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

This paper studies the perturbation of a Lie-Poisson (or, equivalently an Euler-Poincaré) system by a special dissipation term that has Brockett’s double bracket form. We show that a formally unstable equilibrium of the unperturbed system becomes a spectrally and hence nonlinearly unstable equilibrium after the perturbation is added. We also investigate the geometry of this dissipation mechanism and its relation to Rayleigh dissipation functions. This work complements our earlier work (Bloch, Krishnaprasad, Marsden and Ratiu [1991, 1994]) in which we studied the corresponding problem for systems with symmetry with the dissipation added to the internal variables; here it is added directly

Marsden and Scheurle [1993] studied Lagrangian reduction in the context of momentum map constraints—here meaning the reduction of the standard Euler-Lagrange system restricted to a level set of a momentum map. This provides a Lagrangian parallel to the reduction of symplectic manifolds. The present paper studies the Lagrangian parallel of Poisson reduction for Hamiltonian systems. For the reduction of a Lagrangian system on a level set of a conserved quantity, a key object is the Routhian, which is the Lagrangian minus the mechanical connection paired with the fixed value of the momentum map. For unconstrained systems, we use a velocity shifted Lagrangian, which plays the role of the Routhian in the constrained theory. Hamilton’s variational principle for the Euler-Lagrange equations breaks up into two sets of equations that represent a set of Euler-Lagrange equations with gyroscopic forcing that can be written in terms of the curvature of the connection for horizontal variations, and into the Euler-Poincaré equations for the vertical variations. This new set of equations is what we call the reduced Euler-Lagrange equations, and it includes the Euler-Poincaré and the Hamel equations as special cases. We illustrate this methodology for a rigid body with internal rotors and for a particle moving in a magnetic field. 1

Cotangent bundle reduction theory is a basic and well developed subject in which one performs symplectic reduction on cotangent bundles. One starts with a (free and proper) action of a Lie group G on a configuration manifold Q, considers its natural cotangent lift to T ∗ Q and then one seeks realizations of the corresponding symplectic or Poisson reduced space. We further develop this theory by explicitly identifying the symplectic leaves of the Poisson manifold T ∗ Q/G, decomposed as a Whitney sum bundle, T ∗ (Q/G) � �g ∗ over Q/G. The splitting arises naturally from a choice of connection on the G-principal bundle Q → Q/G. The symplectic leaves are computed and a formula for the reduced symplectic form is found.

Reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of Euler, Jacobi, Lagrange, Hamilton, Routh, Poincaré and others. The modern vision of mechanics includes, besides the traditional mechanics of particles and rigid bodies, field theories such as electromagnetism, fluid mechanics, plasma physics, solid mechanics as well as quantum mechanics, and relativistic theories, including gravity. Symmetries in these theories vary from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas, to subtle symmetries underlying integrable systems. Reduction theory concerns the removal of symmetries and their associated conservation laws. Variational principles along with symplectic and Poisson geometry, provide fundamental tools for this endeavor. Reduction theory has been extremely useful in a wide variety of areas, from a deeper understanding of many physical theories, including new variational and Poisson structures, stability theory, integrable systems, as well as geometric phases.

This paper surveys selected recent progress in geometric mechanics, focussing on Lagrangian reduction and gives some new applications to nonholonomic systems, that is, mechanical systems with constraints typified by rolling without slipping. Reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of Euler, Jacobi, Lagrange, Hamilton, Routh, Poincaré and others. The modern vision of mechanics includes, besides the traditional mechanics of particles and rigid bodies, field theories such as electromagnetism, fluid mechanics, plasma physics, solid mechanics as well as quantum mechanics, and relativistic theories, including gravity. Symmetries in mechanics ranges from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas, to subtle symmetries underlying integrable systems. Reduction theory concerns the removal of symmetries and utilizing their associated conservation laws. Reduction theory has been extremely useful in a wide variety of areas, from a deeper understanding of many

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].

We construct a Poisson map between manifolds with linear Poisson brackets corresponding to the two samples of Lie algebra e(3). Using this map we establish equivalence of the Steklov-Lyapunov system and the motion of a particle on the surface of the sphere under the influence of the fourth order potential. To study separation of variables for the Steklov case on the Lie algebra so(4) we use the twisted Poisson map between the bi-Hamiltonian manifolds e(3) and so(4).