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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 physical theories, including new variational and Poisson structures, stability theory, integrable systems, as well as geometric phases. Much effort has gone into the development of the symplectic and Poisson view of reduction theory, but recently the Lagrangian view, emphasizing the reduction of variational principles has also matured. CONTENTS 2While there has been much activity in the geometry of nonholonomic systems, the task of providing an intrinsic geometric formulation of the reduction theory for nonholonomic systems from the point of view of Lagrangian reduction has been somewhat incomplete. One of the purposes of this paper is to finish this task. In particular, we show how to write the reduced Lagrange d'Alembert equations, and in particular, its vertical part, the momentum equation, intrinsically using covariant derivatives. The resulting equations are called the Lagrange-d'Alembert-Poincaré equations. Contents 1 An Introduction to Reduction Theory 2 2 Geometric Mechanics and Nonholonomic Systems 8 3 The Lagrange-d'Alembert Principle with Symmetry 15 4 The Local Momentum and Horizontal Equation 28 5 The Snakeboard 31 6 Miscellany and Future Directions 36 An Introduction to Reduction TheoryThe Purpose of this Paper. This paper outlines some features of general reduction theory as well as the geometry of nonholonomic mechanical systems. In addition to this survey nature, there are some new results. Our previous work on the geometric theory of Lagrangian reduction provides a convenient context that is herein generalized to nonholonomic systems with symmetry. This provides an intrinsic geometric setting for many of the results that were previously understood primarily in coordinates. This solidification and extension of the basic theory should have several interesting consequences, some of which are spelled out in the final section of the paper. Two important references for this work are Cendra, Marsden and Ratiu [2000], hereafter...

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 EulerPoincaré form for right invariant motion on the diffeomorphism group of positionvelocity 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 LiePoisson bracket on the dual of a semidirect product Lie algebra. Because of degeneracies in the Lagrangian, the Legendre transform is dealt with using the Dirac theory of constraints. Another Maxwell-Vlasov Poisson structure is known, whose ingredients are the Lie-Poisson bracket on the dual of the Lie algebra of symplectomorphisms of phase space and the Born-Infeld brackets for the Maxwell field. We discuss the relationship between these two Hamiltonian formulations. We also discuss the general Kelvin-Noether theorem for Euler-Poincaré equations and its meaning in the plasma context.

As is well-known, there is a variational principle for the Euler-Poincaré equations on a Lie algebra g of a Lie group G obtained by reducing Hamilton's principle on G by the action of G by, say, left multiplication. The purpose of this paper is to give a variational principle for the Lie-Poisson equations on g * , the dual of g, and also to generalize this construction.The more general situation is that in which the original configuration space is not a Lie group, but rather a configuration manifold Q on which a Lie group G acts freely and properly, so that Q → Q/G becomes a principal bundle. Starting with a Lagrangian system on T Q invariant under the tangent lifted action of G, the reduced equations on (T Q)/G, appropriately identified, are the Lagrange-Poincaré equations. Similarly, if we start with a Hamiltonian system on T * Q, invariant under the cotangent lifted action of G, the resulting reduced equations on (T * Q)/G are called the Hamilton-Poincaré equations.Amongst our new results, we derive a variational structure for the Hamilton-Poincaré equations, give a formula for the Poisson structure on these reduced spaces that simplifies previous formulas of Montgomery, and give a new representation for the symplectic structure on the associated symplectic leaves. We illustrate the formalism with a simple, but interesting example, that of a rigid body with internal rotors.

The constraint distribution in non-holonomic mechanics has a double role. On one hand, it is a kinematic constraint, that is, it is a restriction on the motion itself. On the other hand, it is also a restriction on the allowed variations when using D'Alembert's Principle to derive the equations of motion. We will show that many systems of physical interest where D'Alembert's Principle does not apply can be conveniently modeled within the general idea of the Principle of Virtual Work by the introduction of both kinematic constraints and variational constraints as being independent entities. This includes, for example, elastic rolling bodies and pneumatic tires. Also, D'Alembert's Principle and Chetaev's Principle fall into this scheme. We emphasize the geometric point of view, avoiding the use of local coordinates, which is the appropriate setting for dealing with questions of global nature, like reduction.