Hamilton–Jacobi equation

Summary

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 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. The qualitative form of this connection is called Hamilton's optico-mechanical analogy. 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. 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 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.

Official source

https://en.wikipedia.org/wiki/Hamilton–Jacobi_equation

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