---
title: "The Euler-Lagrange Equation and Conserved Momenta"
description: "Applying the Euler-Lagrange equation to a Lagrangian returns the equation of motion, and a coordinate missing from the whole Lagrangian gives a conserved momentum when no nonconservative generalized f"
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: 2026-11-12
---

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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 a coordinate missing from the whole Lagrangian gives a conserved momentum when no nonconservative generalized force acts in it. 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 coordinate absent from the whole Lagrangian and writes down the conserved conjugate 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. A dot marks a time derivative: q̇ is the rate of q and q̈ its acceleration. For a coordinate q it reads d/dt(∂L/∂q̇) − ∂L/∂q = 0. 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. With a nonconservative generalised force Q, the right side becomes Q.

A coordinate absent from the whole Lagrangian is called cyclic, and its conjugate momentum ∂L/∂q̇ is conserved. The recipe keeps that derivative constant, so you only need to spot which coordinates the Lagrangian skips. Each such skip is a symmetry of the description. The conservation holds when no nonconservative generalised force acts in that coordinate; with such a force Q, the right side becomes Q and the momentum no longer coasts. Free polar motion with V = 0 has T = ½m(ṙ² + r²φ̇²), which depends on r, so the radial momentum need not be conserved; a straight radial trajectory is the special case where mṙ stays constant.

*(drawing: Differentiate with respect to the rate, take the time derivative, subtract the derivative with respect to the coordinate.)*

**Example.** Pendulum: script L = ½ml²θ̇² + mgl cos θ. Differentiate with respect to the rate: ∂L/∂θ̇ = ml²θ̇, whose time derivative is ml²θ̈. Differentiate with respect to the angle: ∂L/∂θ = −mgl sin θ. Euler-Lagrange sets the difference to zero: ml²θ̈ + mgl sin θ = 0, so θ̈ + (g/l) sin θ = 0. Small swings: sin θ ≈ θ, so θ̈ + (g/l)θ = 0.

In an inertial Cartesian system with the natural L = T − V, if L skips x then no force acts along x and p_x = mẋ coasts unchanged. Rotation symmetry gives conserved angular momentum. A pendulum angle is not cyclic under gravity, so its angular momentum about the pivot is not conserved.

Energy check: dL/dt = Σᵢ(q̇ᵢ ∂L/∂qᵢ + q̈ᵢ ∂L/∂q̇ᵢ) + ∂L/∂t. Replace ∂L/∂qᵢ with d/dt(∂L/∂q̇ᵢ) from the unforced Euler-Lagrange equation, and the sums collapse into d/dt(Σᵢ q̇ᵢ ∂L/∂q̇ᵢ). So d/dt(Σᵢ q̇ᵢ ∂L/∂q̇ᵢ − L) = −∂L/∂t. An autonomous Lagrangian has ∂L/∂t = 0, so the generalized energy E = Σᵢ q̇ᵢ ∂L/∂q̇ᵢ − L is conserved. E = T + V when L = T − V with T homogeneous quadratic in the rates and V rate-independent.

**Recap.** Apply the Euler Lagrange equation to each coordinate, and spot the coordinates the whole Lagrangian skips for conserved conjugate 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)