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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.

Scientists
William Rowan Hamilton (1805-1865)

Recent forum threads on Hamiltonian

Breakdown
Physics 					> Classical Mechanics 					>> Lagrangian/Hamiltonian

See Also
LagrangianHamilton's eqs of motio

Images

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 $T = \frac{p^2}{2m}$

Commentary
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