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What is a Hamiltonian

  1. Jul 24, 2014 #1
    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!
     
  2. jcsd
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