Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 66 / 4 / i / Solution

An isolating integral is a single-valued integral of motion $I(\mathbf{x},\mathbf{v})$ whose fixed value confines an orbit to a lower-dimensional level set of phase space. Along the flow, $dI/dt=0$; for an autonomous Hamiltonian, this means $\{I,H\}=0$. Several isolating integrals are independent when their gradients are linearly independent on the regular region considered. Counting them requires a specified domain: merely local orbit labels in a flow box do not establish global single-valued integrals. The following cases concern regular globally meaningful orbit constraints.