Hamilton's Canonical Equations of Motion

the differential equations of motion of a mechanical system in which the variables are the generalized momenta p_i, as well as the generalized coordinates q_i; the q_i and p_i, are called canonical variables. Hamilton’s canonical equations of motion have the form

where H (q_i, p_i, t) is the Hamiltonian function, which when the constraints are not time-dependent and the acting forces are potential is equal to the sum of the kinetic and potential energies of the system expressed in terms of canonical variables, and s is the number of degrees of freedom of the system. By integrating this system of ordinary first-order differential equations it is possible to find all the q_i and p_i as functions of the time t and 2s constants, which are determined from the initial conditions.

Hamilton’s equations have the important property that they make it possible, using canonical transformations, to transform from the q_i and p_i to new canonical variables Q_i (q_i, p_i, t) and P_i (q_i, p_i, t ) that also satisfy Hamilton’s equations, but with a different function H (Q_i, P_i, t). In this manner, Hamilton’s equations can be reduced to a form that simplifies the process of integration. Hamilton’s equations are used in statistical physics, quantum mechanics, electrodynamics, and other branches of physics, as well as in classical mechanics.