Stationary-action principle

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.

Going deeper

For a student

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.