Global Formulations of Lagrangian and Hamiltonian Dynamics on Embedded Manifolds

Abstract

This paper provides global formulations of Lagrangian and Hamiltonian variational dynamics evolving on a manifold embedded in R n , which appears often in robotics and multi body dynamics. Euler-Lagrange equations and Hamilton's equations are developed in a coordinate-free fashion on a manifold, without relying on local parameterizations that may lead to singularities and cumbersome equations of motion. The proposed intrinsic formulations of Lagrangian and Hamiltonian dynamics are expressed compactly, and they are useful in analysis and computation of the global dynamics. These are illustrated by dynamic systems on the unit-spheres and the special orthogonal group.