nLab Hamilton-Jacobi equation

Idea

The Hamilton-Jacobi equation is a first-order nonlinear partial differential equation that arises in variational calculus and which gives, in particular, a formulation of the laws of motion in classical mechanics.

It relates the principal function (action functional as a function of certain arguments) with the Hamiltonian. In one dimension it is of the form

$$\frac{\partial S}{\partial t} + H\left(t, x, \frac{\partial S}{\partial x}\right) = 0$$

Related concepts

• Hamilton's equations

• prequantized Lagrangian correspondence

• Bohmian mechanics

References

• eom: Hamilton-Jacobi theory
• wikipedia, scholarpedia
• V.I. Arnol’d, Mathematical methods of classical mechanics
• Shlomo Sternberg, Lectures in differential geometry
• L. Vitagliano, Hamilton-Jacobi diffieties, arxiv/1104.0162

Specifically for quadratic Hamiltonians:

• Joel Robbin, Dietmar Salamon, remark 3.4 in Feynman path integrals on phase space and the metaplectic representation in Dietmar Salamon (ed.), Symplectic Geometry, LMS Lecture Note series 192 (1993) (pdf)