the Hamiltonian
/ HAM-il-tun /
The Hamiltonian is the single function that runs the clockwork of classical mechanics in phase space. Where the Lagrangian is built from the difference of kinetic and potential energy and lives on positions and velocities, the Hamiltonian is (usually) the total energy of the system written in terms of positions and momenta. Give me H, and I can tell you how every coordinate and every momentum evolves in time.
Formally, H(q, p, t) is the Legendre transform of the Lagrangian in the velocities: H = sum_i p_i q_dot_i - L, with p_i = partial L / partial q_dot_i the conjugate momenta and every q_dot eliminated in favour of p. For a wide class of systems -- those where the kinetic energy is quadratic in the velocities and the constraints do not depend on time -- H equals the total energy T + V. The variables it depends on matter: H is a function on phase space (of q and p), not of velocity, and that is exactly what makes it the generator of time evolution through Hamilton's equations, q_dot = partial H / partial p and p_dot = - partial H / partial q.
The Hamiltonian is the pivot between classical and quantum physics. In quantum mechanics the classical H becomes the Hamiltonian operator, whose eigenvalues are the allowed energies and which generates time evolution through the Schrodinger equation, i hbar partial psi / partial t = H psi. In statistical mechanics the same H sets the Boltzmann factor exp(-H / k_B T) that weights every microstate. Almost every advanced theory is specified by writing down its Hamiltonian (or its cousin, the Lagrangian density).
A particle of mass m in a potential V(x) has L = (1/2) m x_dot^2 - V(x), momentum p = m x_dot, and Hamiltonian H = p^2/(2m) + V(x). For a mass on a spring, V = (1/2) k x^2, so H = p^2/(2m) + (1/2) k x^2 -- kinetic plus potential energy, now the master function of the motion.
The harmonic-oscillator Hamiltonian: total energy expressed in (x, p).
'Hamiltonian = energy' is a frequent trap. The equality holds only when the kinetic energy is a quadratic form in the velocities and the transformation to generalized coordinates carries no explicit time dependence; whether H is conserved is a separate question, governed by whether partial H / partial t = 0.