Hamiltonian: Definition, Equations & Explanation

[Greg Bernhardt]

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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[AaronS]

Thanks for sharing this useful information! I didn't know that the Hamiltonian is a function which summarizes equations of motion and gives the total energy of a system. It's great to learn the equations and extended explanation to understand it better.