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 or 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
In a simple system, i.e., single Particle traveling on a single dimension:
ℋ = T + V where p² T = —— 2m V = V(q)