---
title: "Hamilton's Principle and the Action"
description: "The paths a system can actually follow between fixed endpoints and times are those that make the action stationary, and the Euler-Lagrange equations are the condition for that. Stating mechanics as a "
canonical: https://lightmysky.com/learn/science/hamiltons-principle-and-the-action-mt_a273REeeGu
source: https://lightmysky.com/learn/science/hamiltons-principle-and-the-action-mt_a273REeeGu.md
retrieved: 2026-11-12
---

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# Hamilton's Principle and the Action

The paths a system can actually follow between fixed endpoints and times are those that make the action stationary, and the Euler-Lagrange equations are the condition for that. Stating mechanics as a claim about whole paths rather than about instants is what carries over to fields and to quantum theory.

Subject: Science · Area: Forces & Motion · Ages 22 to 23
Page: https://lightmysky.com/learn/science/hamiltons-principle-and-the-action-mt_a273REeeGu

## Ready when they can

- States Hamilton's principle and writes the action of a simple system
- Shows that making the action stationary returns the equation of motion
- Explains what varying a path means and which endpoints stay fixed

## Lesson: The path that stands still

The action S is one number for a whole path: S = ∫_{t1}^{t2} L dt, with L the Lagrangian along the path. Hamilton's principle states that the path a system follows between fixed start and end positions at fixed times t1 and t2 makes S stationary: every small bend keeping those endpoints fixed changes S by nothing to first order. Stationary means the first variation of S vanishes for all such admissible bends, not that S is a minimum; a saddle can qualify.

**Example.** Worked example: one coordinate x with L = ½mẋ² − V(x), and endpoints x(t1) = x1, x(t2) = x2 fixed. Bend the path: x(t) + η(t), with η small and η(t1) = η(t2) = 0. The first change in S is ∫(∂L/∂x η + ∂L/∂ẋ η̇) dt. Integrate the second term by parts: the boundary term (∂L/∂ẋ η) at t1 and t2 vanishes because η is zero there. What remains is ∫(∂L/∂x − d/dt(∂L/∂ẋ)) η dt. Since η is arbitrary, the bracket vanishes: the Euler-Lagrange equation. Free particle check: V = 0 gives mẍ = 0, a straight line at constant speed.

*(drawing: Bend the path with pinned ends, integrate by parts, and the Euler-Lagrange bracket remains.)*

Varying a path means bending it while pinning both endpoints: same start position and time, same end position and time. The bends are all smooth paths with those pinned ends, and stationarity demands zero first variation under every one of them, not under a single chosen nudge. This fixed-endpoint version is the one that returns the interior Euler-Lagrange equation directly. Leaving an end or end time free adds boundary terms; those become extra conditions on the free end, while the interior equation still follows from bends that vanish near the ends.

Stationary does not mean least: for ordinary positive kinetic energy the action can sit at a saddle, and several different paths can be stationary at once, so fixed endpoints need not pick a unique trajectory.

Mechanics becomes a claim about whole paths rather than instants, and that pattern travels: fields vary whole histories, and quantum theory weighs all paths at once. The fixed endpoint times are part of the setup, not an optimization over shorter times. Learn the stationary habit once and reuse it everywhere.

**Recap.** Pin positions and times at both ends, demand zero first variation under all bends, and the equation of motion drops out.

## Practice

8 questions on this page, each with its working shown.

## Needs first

- [The Euler-Lagrange Equation and Conserved Momenta](https://lightmysky.com/learn/science/the-euler-lagrange-equation-and-conserved-momenta-mt_GSEFABbEXI)

## Optional connections

- [Generalised Coordinates and the Lagrangian](https://lightmysky.com/learn/science/generalised-coordinates-and-the-lagrangian-mt_9QPk1zI9jE)

## Opens up

- [The Hamiltonian, Phase Space and the Canonical Equations](https://lightmysky.com/learn/science/the-hamiltonian-phase-space-and-the-canonical-equations-mt_9bMDXQiq_n)
- [The Path Integral and the Sum Over Histories](https://lightmysky.com/learn/science/the-path-integral-and-the-sum-over-histories-mt_QRLheflB1O)