Hamiltonian mechanics6 is based on the Hamiltonian function H(p, q). representing the total energy in terms of the generalized coordinate p and generalized momentum q. (Newtonian mechanics focuses on forces.) The Hamiltonian system is then defined by

For the undamped mass-spring system with mass m, spring constantk, and displacement x, we let q = x and p = mx (the momentum).
(a) Show that the kinetic energy of the mass is p2 / 2m.
(b) Show that the total energy is H (p, q) = p2/2m + kq2/2.
(c) Derive the corresponding Hamiltonian system.

aH q=_ ap a H and p=