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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)

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