In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics. The Hamilton–Jacobi equation is particularly useful in identifying conserved quantities for mechanical systems, which may be possible even when the mechanical problem itself cannot be solved completely. The Hamilton–Jacobi equation is also the only formulation of mechanics in which the motion of a particle can be represented as a wave. In this sense, it fulfilled a long-held goal of theoretical physics (dating at least to Johann Bernoulli in the eighteenth century) of finding an analogy between the propagation of light and the motion of a particle. The wave equation followed by mechanical systems is similar to, but not identical with, Schrödinger's equation, as described below; for this reason, the Hamilton–Jacobi equation is considered the "closest approach" of classical mechanics to quantum mechanics. In mathematics, the Hamilton–Jacobi equation is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations. It can be understood as a special case of the Hamilton–Jacobi–Bellman equation from dynamic programming. Notation Boldface variables such as represent a list of generalized coordinates, A dot over a variable or list signifies the time derivative (see Newton's notation). For example, The dot product notation between two lists of the same number of coordinates is a shorthand for the sum of the products of corresponding components, such as Hamilton's principal function Definition Let the Hessian matrix be invertible. The relation shows that the Euler–Lagrange equations form a system of second-order ordinary differential equations. Inverting the matrix transforms this system into Let a time instant and a point in the configuration space be fixed. The existence and uniqueness theorems guarantee that, for every the initial value problem with the conditions and has a locally unique solution Additionally, let there be a sufficiently small time interval such that extremals with different initial velocities would not intersect in The latter means that, for any and any there can be at most one extremal for which and Substituting into the action functional results in the Hamilton's principal function (HPF) where Formula for the momenta: pi(q,t) = ∂S/∂qi The momenta are defined as the quantities This section shows that the dependency of on disappears, once the HPF is known. Indeed, let a time instant