Direct product of integrable Hamiltonian systems
= Direct product of integrable Hamiltonian systems
The direct product of integrable Hamiltonian systems is integrable. Pull each family's commuting <integrals of motion> back to the product <phase space>. Integrals from different factors have zero <Poisson bracket> because they use disjoint canonical coordinates, and their <total differential>[differentials] are independent wherever the differentials within each factor are independent.