# nLab Hamilton-Jacobi equation

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## Surveys, textbooks and lecture notes

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• Axiomatizations

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# Contents

## 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

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