Global Formulations of Lagrangian and Hamiltonian Mechanics on Two-Spheres

TL;DR

This paper develops coordinate-free, intrinsic formulations of Lagrangian and Hamiltonian mechanics on products of two-spheres, enabling global analysis without local parameterization singularities.

Contribution

It introduces novel intrinsic formulations of variational dynamics on two-spheres, avoiding local parameterization issues and providing compact, geometric equations of motion.

Findings

Four types of Euler-Lagrange equations derived

Hamilton's equations formulated intrinsically

Equations facilitate global analysis and computation

Abstract

This paper provides global formulations of Lagrangian and Hamiltonian variational dynamics evolving on the product of an arbitrary number of two-spheres. Four types of Euler-Lagrange equations and Hamilton's equations are developed in a coordinate-free fashion on two-spheres, 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 novel in that they incorporate the geometry of two-spheres, resulting in equations of motion that are expressed compactly, and they are useful in analysis and computation of the global dynamics.