  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.
