Show that in a system with n degrees of freedom there exist at most n functionally independent integrals of motion. (c) Show that in the hypothesis of Liouville’s theorem for integrable systems there...

Show that in a system with n degrees of freedom there exist at most n functionally independent integrals of motion. (c) Show that in the hypothesis of Liouville’s theorem for integrable systems there are n tangent vectors tangent to the torus and construct them. Show that the involution property of the integrals of motion implies the local existence of S such that ∇S = p. (d) Consider an n− 1 parameter family of trajectories in an integrable system. Show that the envelopes of the family coincide with caustics.