---
tag: 6JGM1K39FB
kind: concept
name: Hamilton–Jacobi theory of the calculus of variations
msc: [49L99, 70H20, 49K05]
sources:
  - url: https://en.wikipedia.org/wiki/Hamilton–Jacobi_equation
---

## Summary

Hamilton–Jacobi theory links a problem of the [[calculus of variations|PQ25NHV0KS]] with a single first-order partial differential equation, the [[Hamilton–Jacobi equation|5TRXQKP7DY]].
A solution $S$ of this equation, such as the action along extremals, generates a whole family of extremals, a [[field of extremals|K4DFT3H9BJ]].
Conversely, a suitable family of solutions, a complete integral, gives all solutions of Hamilton's canonical equations by Jacobi's theorem.
The theory gives sufficient conditions for minima and is the classical ancestor of dynamic programming.

## Explanations

### L1 human

In a variational problem, one looks for the curve that makes some cost as small as possible, such as the travel time of light.
Instead of following one curve, Hamilton–Jacobi theory looks at the least cost to reach every point, like wave fronts spreading out from a source.
The curves of least cost cross these fronts in a definite direction, as light rays cross the wave fronts.
So knowing the fronts, which solve one partial differential equation, gives all the best curves at once.