---
title: "The Euler-Lagrange Equation and Conserved Momenta"
description: "Applying the Euler-Lagrange equation to a Lagrangian returns the equation of motion, and any coordinate missing from the Lagrangian gives a conserved momentum straight away. Conservation laws stop bei"
canonical: https://lightmysky.com/learn/science/the-euler-lagrange-equation-and-conserved-momenta-mt_GSEFABbEXI
source: https://lightmysky.com/learn/science/the-euler-lagrange-equation-and-conserved-momenta-mt_GSEFABbEXI.md
retrieved: 2026-09-12
---

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# The Euler-Lagrange Equation and Conserved Momenta

Applying the Euler-Lagrange equation to a Lagrangian returns the equation of motion, and any coordinate missing from the Lagrangian gives a conserved momentum straight away. Conservation laws stop being separate rules and become consequences of symmetry.

Subject: Science · Area: Forces & Motion · Ages 21 to 22
Page: https://lightmysky.com/learn/science/the-euler-lagrange-equation-and-conserved-momenta-mt_GSEFABbEXI

## Ready when they can

- Applies the Euler-Lagrange equation to recover a known equation of motion
- Spots a cyclic coordinate and writes down the conserved momentum it implies
- Connects a symmetry of the Lagrangian to the quantity it conserves

## Lesson: From energy to motion

The Euler Lagrange equation turns a Lagrangian into equations of motion, one per generalised coordinate. You differentiate L with respect to the rate, take the time derivative, and subtract the derivative with respect to the coordinate itself. Applied to a falling mass or a pendulum it reproduces the Newton result, which is the first consistency check to run.

A coordinate missing from L is called cyclic, and it hands you a conserved momentum at once: the derivative of L with respect to its rate stays constant. The recipe does the bookkeeping, so you only need to spot which coordinates the Lagrangian skips. Each such skip is a symmetry of the description.

**Example.** Take the pendulum Lagrangian in the angle theta. The recipe returns the familiar swing equation with angular acceleration plus g over L times sin theta equals zero. For small swings the sine straightens and you recover the small angle oscillator, matching what Newton gives.

**Tip.** Read each conservation law as a symmetry consequence. When the potential skips some direction there is no force along it, so motion in that direction coasts unchanged. The classic payoff is rotation symmetry giving conserved angular momentum, just as shifting the whole clock gives energy conservation.

**Recap.** Apply the recipe for motion, spot the missing coordinates for conserved momenta.

## 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)
- [Partial Derivatives](https://lightmysky.com/learn/mathematics/partial-derivatives-mt_tu11fd9Xk9)

## Opens up

- [Hamilton's Principle and the Action](https://lightmysky.com/learn/science/hamiltons-principle-and-the-action-mt_a273REeeGu)