What is a Hamiltonian

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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 [itex]p_a(t)[/itex] for each canonical coordinate [itex]q_a(t)[/itex]:
[itex]p_a = \frac{\partial L}{\partial \dot q_a}[/itex]

The Hamiltonian is given by
[itex]\left(\sum_a p_a \dot q_a \right) - L[/itex]

Hamilton's equations of motion are
[itex]\dot q_a = \frac{\partial H}{\partial p_a}[/itex]
[itex]\dot p_a = - \frac{\partial H}{\partial q_a}[/itex]

The Hamiltonian has the interesting property that
[itex]\dot H = \frac{\partial H}{\partial t}[/itex]

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):
[itex]L = T - V[/itex]

where
[itex]T = \frac12 m \left( \frac{dq}{dt} \right)^2[/itex]

For canonical coordinate q, we find canonical momentum p:
[itex]p = m \frac{dq}{dt}[/itex]

and from that, we find the Hamiltonian:
[itex]H = T + V[/itex]

where the kinetic energy is now given by
[itex]T = \frac{p^2}{2m}[/itex]

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
 

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