# What is a Hamiltonian

Definition/Summary

The Hamiltonian, like the Lagrangian, is a function that summarizes equations of motion. It has the additional interpretation of giving the total energy of a system.

Though originally stated for classical mechanics, it is also an important part of quantum mechanics.

Equations

Start from the Lagrangian and define a canonical momentum $p_a(t)$ for each canonical coordinate $q_a(t)$:
$p_a = \frac{\partial L}{\partial \dot q_a}$

The Hamiltonian is given by
$\left(\sum_a p_a \dot q_a \right) - L$

Hamilton's equations of motion are
$\dot q_a = \frac{\partial H}{\partial p_a}$
$\dot p_a = - \frac{\partial H}{\partial q_a}$

The Hamiltonian has the interesting property that
$\dot H = \frac{\partial H}{\partial t}$

meaning that if the Hamiltonian has no explicit time dependence, it is a constant of the motion.

Extended explanation

To illustrate the derivation of the Hamiltonian, let us start with the Lagrangian for a particle with Newtonian kinetic energy and potential energy V(q):
$L = T - V$

where
$T = \frac12 m \left( \frac{dq}{dt} \right)^2$

For canonical coordinate q, we find canonical momentum p:
$p = m \frac{dq}{dt}$

and from that, we find the Hamiltonian:
$H = T + V$

where the kinetic energy is now given by
$T = \frac{p^2}{2m}$

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