 Hamilton's equations are

where the Hamiltonian

is a function of the configuration and conjugate ""generalized" momenta"

in which is a shorthand notation for a vector of partial derivatives with respect to the indicated variables (see for example matrix calculus for this denominator notation), and possibly time ,

Setting up the Hamiltonian of the system, then substituting into the equations and evaluating the partial derivatives and simplifying, a set of coupled first order ODEs in the coordinates and momenta are obtained.The Hamilton–Jacobi equation is

where

is "Hamilton's principal function", also called the "classical action" is a functional of .In this case, the momenta are given by

Although the equation has a simple general form, for a given Hamiltonian it is actually a single first order "non-linear" PDE, in variables.