Hamiltonian

Hamiltonian (or Hamiltonian function) refers to a type of function used in Hamiltonian mechanics, the function yielding a closed system's (i.e., one in which energy is not at the time being exchanged with the outside world) total energy: its kinetic energy plus its potential energy. The function is key to Hamiltonian mechanics, which uses two equations (the Hamilton equations of motion) that relate the Hamiltonian function, ℋ(q,p,t), to three variables it depends on: space coordinates q, momentum coordinates p and time t:

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)

• T - kinetic energy.
• m - mass of particle.
• V - potential energy, a function of space-coordinates q.

Hamiltonian mechanics was devised to solve classical mechanical problems and later proved useful in statistical mechanics and quantum mechanics.