The law of conservation of energy states that the energy in a Hamiltonian system is constant on solutions.
(a) Prove that if u(t) satisfies the Hamiltonian system (9.22), then H(u(t)) ≡ c is a constant, and hence solutions u(t) move along the level sets of the Hamiltonian or energy function. Explain how the value of c is determined by the initial conditions.
(b) Plot the level curves of the particular Hamiltonian function H(u. v) = u2 - 2uv + 2v2 and verify that they coincide with the solution trajectories.