Hamiltonian (or Hamiltonian function) is the name of a function used in Hamiltonian mechanics consisting of (in a closed system), the total energy of the system, consisting of kinetic energy plus potential energy. Hamiltonian mechanics can solve classical mechanical problems and was useful as a step toward statistical mechanics and quantum mechanics.
The Hamiltonian function, ℋ(q,p,t), of space coordinates q, momentum coordinates p and time t satisfies:
dp ∂ℋ —— = - —— dt ∂q dq ∂ℋ —— = + —— dt ∂p
(These are derived from Newton's laws.) In a simple system, i.e., single particle traveling on a single dimension:
ℋ = T + V where p² T = —— 2m V = V(q)