In mathematics, the Hamilton–Jacobi equation (HJE) is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations, and is a special case of the Hamilton–Jacobi–Bellman equation. It is named for William Rowan Hamilton and Carl Gustav Jacob Jacobi. In physics, it is a 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 HJE is also the only formulation of mechanics in which the motion of a particle can be represented as a wave. In this sense, the HJE fulfilled a long-held goal of theoretical physics (dating at least to Johann Bernoulli in the 18th 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 HJE is considered the "closest approach" of classical mechanics to quantum mechanics.
In Riemannian geometry, a Jacobi field is a vector field along a geodesic in a Riemannian manifold describing the difference between the geodesic and an "infinitesimally close" geodesic. In other words, the Jacobi fields along a geodesic form the tangent space to the geodesic in the space of all geodesics. They are named after Carl Jacobi.
Jacobi fields can be obtained in the following way: Take a smooth one parameter family of geodesics with
, then
is a Jacobi field, and describes the behavior of the geodesics in an infinitesimal neighborhood of a
given geodesic .
A vector field J along a geodesic is said to be a Jacobi field if it satisfies the Jacobi equation:
where D denotes the covariant derivative with respect to the Levi-Civita connection, R the Riemann curvature tensor, the tangent vector field, and t is the parameter of the geodesic.
On a complete Riemannian manifold, for any Jacobi field there is a family of geodesics
describing the field (as in the preceding paragraph).