---
title: "Hamilton's Principle and the Action"
description: "The path a system actually follows is the one that makes the action stationary, and the Euler-Lagrange equations are the condition for that. Stating mechanics as a claim about whole paths rather than "
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: 2026-09-12
---

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# Hamilton's Principle and the Action

The path a system actually follows is the one that makes 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: the time integral of kinetic minus potential energy along it. Hamilton principle says the path nature takes makes S stationary, meaning small bends of the path change S by nothing to first order.

**Example.** Picture a tossed ball: the true arc and a slightly higher arc between the same throws. Nudge the true arc and the action barely shifts; nudge the wrong arc and it shifts at once. Stationarity picks the thrown path out of all imagined ones.

Varying a path means bending it while pinning both endpoints: same start position and time, same end position and time. Demanding stationarity under every such bend returns the Euler-Lagrange equation, which is the equation of motion in disguise.

**Tip.** 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. Learn the stationary habit once and reuse it everywhere.

**Recap.** Pin the endpoints, stationize the action, and the equation of motion drops out.

## Practice

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

## Needs first

- [Generalised Coordinates and the Lagrangian](https://lightmysky.com/learn/science/generalised-coordinates-and-the-lagrangian-mt_9QPk1zI9jE)
- [The Euler-Lagrange Equation and Conserved Momenta](https://lightmysky.com/learn/science/the-euler-lagrange-equation-and-conserved-momenta-mt_GSEFABbEXI)

## 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)