Symmetry reduction of discrete Lagrangian mechanics on Lie groups

Abstract

For a discrete mechanical system on a Lie group $G$ determined by a (reduced) Lagrangian $\ell$ we define a Poisson structure via the pull-back of the Lie-Poisson structure on the dual of the Lie algebra ${\mathfrak{g}}^{\ast }$ 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 ${\omega }_{\mathbb{L}}$ on $G\times G$. 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.

References in corpus (1)

• Discrete Euler-Poincaré and Lie-Poisson Equations

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• Lagrangian reduction of forced discrete mechanical systems
• Conservation laws of semidiscrete canonical Hamiltonian equations
• Discrete Formulation for the dynamics of rods deforming in space
• Gauge reduction in covariant field theory