and a point in the configuration space be fixed. For every time instant and a point let be the (unique) extremal from the definition of the Hamilton's principal function Call the velocity at . Then Mathematical formulation Given the Hamiltonian of a mechanical system, the Hamilton–Jacobi equation is a first-order, non-linear partial differential equation for the Hamilton's principal function , Alternatively, as described below, the Hamilton–Jacobi equation may be derived from Hamiltonian mechanics by treating as the generating function for a canonical transformation of the classical Hamiltonian The conjugate momenta correspond to the first derivatives of with respect to the generalized coordinates As a solution to the Hamilton–Jacobi equation, the principal function contains undetermined constants, the first of them denoted as , and the last one coming from the integration of . The relationship between and then describes the orbit in phase space in terms of these constants of motion. Furthermore, the quantities are also constants of motion, and these equations can be inverted to find as a function of all the and constants and time. Comparison with other formulations of mechanics The Hamilton–Jacobi equation is a single, first-order partial differential equation for the function of the generalized coordinates and the time . The generalized momenta do not appear, except as derivatives of . Remarkably, the function is equal to the classical action. For comparison, in the equivalent Euler–Lagrange equations of motion of Lagrangian mechanics, the conjugate momenta also do not appear; however, those equations are a system of , generally second-order equations for the time evolution of the generalized coordinates. Similarly, Hamilton's equations of motion are another system of 2N first-order equations for the time evolution of the generalized coordinates and their conjugate momenta . Since the HJE is an equivalent expression of an integral minimization problem such as Hamilton's principle, the HJE can be useful in other problems of the calculus of variations and, more generally, in other branches of mathematics and physics, such as dynamical systems, symplectic geometry and quantum chaos. For example, the Hamilton–Jacobi equations can be used to determine the geodesics on a Riemannian manifold, an important variational problem in Riemannian geometry. Derivation using a canonical transformation Any canonical transformation involving a type-2 generating function leads to the relations and Hamilton's equations in terms of the new variables and new Hamiltonian have the same form: To derive the HJE, a generating function is chosen in