The Hamilton–Jacobi equation is a 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.
[1][2] 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.
A dot over a variable or list signifies the time derivative (see Newton's notation).
is called perturbation, infinitesimal variation or virtual displacement of the mechanical system at the point
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
then describes the orbit in phase space in terms of these constants of motion.
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.
Ideally, these N equations can be inverted to find the original generalized coordinates
When the problem allows additive separation of variables, the HJE leads directly to constants of motion.
is sometimes called the abbreviated action or Hamilton's characteristic function [5]: 434 and sometimes[9]: 607 written
Substitution of these formulae into the Hamilton–Jacobi equation shows that the function ψ must be a constant (denoted here as
The separability of S depends both on the Hamiltonian and on the choice of generalized coordinates.
For orthogonal coordinates and Hamiltonians that have no time dependence and are quadratic in the generalized momenta,
will be completely separable if the potential energy is additively separable in each coordinate, where the potential energy term for each coordinate is multiplied by the coordinate-dependent factor in the corresponding momentum term of the Hamiltonian (the Staeckel conditions).
In spherical coordinates the Hamiltonian of a free particle moving in a conservative potential U can be written
The Hamilton–Jacobi equation is completely separable in these coordinates provided that there exist functions
dependence and reduces the HJE to the final ordinary differential equation
yields the reduced Hamilton–Jacobi equation (after re-arrangement and multiplication of both sides by the denominator)
yields the reduced Hamilton–Jacobi equation (after re-arrangement and multiplication of both sides by the denominator)
[10] For example, in geometrical optics, light can be considered either as “rays” or waves.
More precisely, geometrical optics is a variational problem where the “action” is the travel time
-isosurface as a function of time is defined by the motions of the particles beginning at the points
in vacuum, the Hamilton–Jacobi equation in geometry determined by the metric tensor
, implying the particle moving along a circular trajectory with a permanent radius
For the flat, monochromatic, linearly polarized wave with a field
implying the particle figure-8 trajectory with a long its axis oriented along the electric field
plane set perpendicular to the solenoid axis with arbitrary azimuth angle