---
tag: Q104C67P26
kind: statement
role: fact
name: Stationary-action principle
msc: [49S05, 70H30, 70H03]
uses: [1EQM0DJ4NC, 5B8NFYB2XQ, PS0VV149ET]
about: [PS0VV149ET]
sources:
  - url: https://en.wikipedia.org/wiki/Action_principles
---

## Statement

Let a physical system be described by generalized coordinates $q = (q_1, \dots, q_n)$ in an open set $Q \subseteq \R^n$, and by a Lagrangian $L \colon [t_0, t_1] \times Q \times \R^n \to \R$ of class $C^2$.
For a $C^2$ path $q \colon [t_0, t_1] \to Q$, its *action* is the [[functional|1EQM0DJ4NC]]
$$S[q] = \int_{t_0}^{t_1} L\big(t, q(t), \dot q(t)\big) \, dt.$$
Fix endpoints $q(t_0) = a$ and $q(t_1) = b$.
The principle states that the motions of the system between them are exactly the $C^2$ paths $q$ with these endpoints that satisfy one condition.
The [[first variation|5B8NFYB2XQ]] of $S$ at $q$ vanishes: $\frac{d}{d\varepsilon} S[q + \varepsilon h] \big|_{\varepsilon = 0} = 0$ for every $C^2$ map $h$ with $h(t_0) = h(t_1) = 0$.
Equivalently, the motions are the solutions of the [[Euler–Lagrange equations|PS0VV149ET]]
$$\frac{d}{dt} \frac{\partial L}{\partial \dot q_i} = \frac{\partial L}{\partial q_i}, \qquad i = 1, \dots, n.$$

## Remarks

The equivalence of the two forms is a theorem of the calculus of variations.
Differentiating under the integral and integrating by parts gives $\frac{d}{d\varepsilon} S[q + \varepsilon h] \big|_{\varepsilon = 0} = \int_{t_0}^{t_1} \sum_i \left( \frac{\partial L}{\partial q_i} - \frac{d}{dt} \frac{\partial L}{\partial \dot q_i} \right) h_i \, dt$.
By the fundamental lemma of the calculus of variations, this vanishes for all $h$ exactly when the Euler–Lagrange equations hold.
The physical content is the choice of $L$; for a particle of mass $m$ in a potential $V$, the choice $L = \frac12 m \abs{\dot q}^2 - V(q)$ gives Newton's law $m \ddot q = -\nabla V(q)$.
The principle asks for a stationary action, not a least one.
For the harmonic oscillator $L = \frac12 (\dot q^2 - \omega^2 q^2)$ with $\omega > 0$, the solution is a minimum of $S$ when $t_1 - t_0 < \pi / \omega$, but not when $t_1 - t_0 > \pi / \omega$: then $h(t) = \sin(\pi (t - t_0)/(t_1 - t_0))$ makes the second variation $\int (\dot h^2 - \omega^2 h^2) \, dt$ negative.
The same principle, with an action given by the integral of a Lagrangian density over space-time, gives the field equations of electromagnetism and of general relativity.
It is also the setting of Noether's theorem on symmetries and conservation laws.
Hamilton's principle in mechanics and Maupertuis's principle are particular forms of it.

## Explanations

### L1 human

Nature seems to choose its motions by an economy rule.
Consider all the ways a system could go from one state to another in a given time.
It takes one for which a single number, the action, does not change to first order when the path is bent slightly.
From this one rule, with the right choice of Lagrangian, come the equations of mechanics, optics, electromagnetism and gravitation.

### L2 human

In practice the principle is a recipe: write the kinetic and potential energies in any convenient coordinates, form $L$, and read off the equations of motion from the Euler–Lagrange equations.
This avoids computing constraint forces, and it makes symmetries visible, since a symmetry of $L$ gives a conserved quantity by Noether's theorem.
The Legendre transform of $L$ leads to Hamilton's equations, and the action itself, as a function of the endpoints, solves the Hamilton–Jacobi equation.