What are the Hamilton equations of motion

1. Jul 24, 2014

Greg Bernhardt

Definition/Summary

Hamilton's equations of motion is a very general equation of a system evolving deterministically in phase space.

Equations

$$\left( {\begin{array}{*{20}{c}} {\dot q}\\ {\dot p} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 0&1\\ { - 1}&0 \end{array}} \right)\left( {\begin{array}{*{20}{c}} {\partial H/\partial q}\\ {\partial H/\partial p} \end{array}} \right)$$

Extended explanation

A common misconception is that Hamilton's equations is only applicable in classical mechanics (celestial mechanics and elastic mechanics). Although this may have once been true, it has found applications everywhere in physics. It applies in almost every conservative system in classical (deterministic) physics. The meat of Hamilton's equations is given by the function H (the 'Hamiltonian'). This is a real function given by position variables and momenta variables with continuous second partial derivatives.

In Newtonian gravity, the Hamiltonian may be given as
$$H = \frac{{{p^2}}}{{2}} - \phi$$
Here, phi is the gravitational potential. Mass is taken to be unity.

In classical electrodynamics, the Hamiltonian may be given as
$$H = \frac{1}{2}{\left( {{\bf{p}} - e{\bf{A}}} \right)^2} + e\phi$$
Here, e is electric charge, phi is now the electric potential, and A is the magnetic vector potential. Mass is again taken to be unity.

In general relativity, the expression of the Hamiltonian function gets far more complicated.

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