Abstract. In this paper, discrete analogues of Euler-Poincaré and Lie-Poisson reduction theory are developed for systems on finite dimensional Lie groups G with Lagrangians L : T G → R that are G-invariant. These discrete equations provide 'reduced' numerical algorithms which manifestly preserve the symplectic structure. The manifold G × G is used as an approximation of T G, and a discrete Langragian L : G × G → R is constructed in such a way that the G-invariance property is preserved. Reduction by G results in a new 'variational' principle for the reduced Lagrangian : G → R, and provides the discrete Euler-Poincaré (DEP) equations. Reconstruction of these equations recovers the discrete Euler-Lagrange equations developed by Marsden et al (Marsden J E, Patrick G and Shkoller S 1998 Commun. Math. Phys. 199 351-395) and Wendlandt and Marsden (Wendlandt J M and Marsden J E 1997 Physica D 106 223-246) which are naturally symplecticmomentum algorithms. Furthermore, the solution of the DEP algorithm immediately leads to a discrete Lie-Poisson (DLP) algorithm. It is shown that when G = SO(n), the DEP and DLP algorithms for a particular choice of the discrete Lagrangian L are equivalent to the Moser-Veselov scheme for the generalized rigid body.

This paper presents a variational and multisymplectic formulation of both compressible and incompressible models of continuum mechanics on general Riemannian manifolds. A general formalism is developed for non-relativistic first-order multisymplectic field theories with constraints, such as the incompressibility constraint. The results obtained in this paper set the stage for multisymplectic reduction and for the further development of Veselov-type multisymplectic discretizations and numerical algorithms. The latter will be the subject of a companion paper

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.

In this paper we analyze the dynamics of N point vortices moving on a sphere from the point of view of geometric mechanics. The formalism is developed for the general case of N vortices, and the details are worked out for the ͑integrable͒ case of three vortices. The system under consideration is SO͑3͒ invariant; the associated momentum map generated by this SO͑3͒ symmetry is equivariant and corresponds to the moment of vorticity. Poisson reduction corresponding to this symmetry is performed; the quotient space is constructed and its Poisson bracket structure and symplectic leaves are found explicitly. The stability of relative equilibria is analyzed by the energy-momentum method. Explicit criteria for stability of different configurations with generic and nongeneric momenta are obtained. In each case a group of transformations is specified, modulo which one has stability in the original ͑unreduced͒ phase space. Special attention is given to the distinction between the cases when the relative equilibrium is a nongreat circle equilateral triangle and when the vortices line up on a great circle.

Abstract. For a discrete mechanical system on a Lie group G determined by a (reduced) Lagrangian ℓ we define a Poisson structure via the pull-back of the Lie-Poisson structure on the dual of the Lie algebra g * by the corresponding Legendre transform. The main result shown in this paper is that this structure coincides with the reduction under the symmetry group G of the canonical discrete Lagrange 2-form ω L on G × G. Its symplectic leaves then become dynamically invariant manifolds for the reduced discrete system. Links between our approach and that of groupoids and algebroids as well as the reduced Hamilton-Jacobi equation are made. The rigid body is discussed as an example.