---
title: "Generalised Coordinates and the Lagrangian"
description: "Choosing coordinates that already respect the constraints removes the constraint forces from the problem, and the Lagrangian is kinetic minus potential energy written in those coordinates. A pendulum "
canonical: https://lightmysky.com/learn/science/generalised-coordinates-and-the-lagrangian-mt_9QPk1zI9jE
source: https://lightmysky.com/learn/science/generalised-coordinates-and-the-lagrangian-mt_9QPk1zI9jE.md
retrieved: 2026-09-12
---

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# Generalised Coordinates and the Lagrangian

Choosing coordinates that already respect the constraints removes the constraint forces from the problem, and the Lagrangian is kinetic minus potential energy written in those coordinates. A pendulum needs one angle rather than two coordinates and a tension.

Subject: Science · Area: Forces & Motion · Ages 21 to 22
Page: https://lightmysky.com/learn/science/generalised-coordinates-and-the-lagrangian-mt_9QPk1zI9jE

## Ready when they can

- Picks a set of generalised coordinates for a constrained system and counts the degrees of freedom
- Writes kinetic and potential energy in those coordinates and forms the Lagrangian
- Explains why constraint forces do not appear once the coordinates respect the constraint

## Lesson: Energy instead of forces

Rolling needed three equations and gyroscopes needed torque bookkeeping. The Lagrangian method sidesteps that grind: it rewrites mechanics around energy instead of forces. You pick generalised coordinates that already respect the constraints, write kinetic minus potential energy in them, and fixed recipes produce the motion. Counting degrees of freedom first tells you exactly how many coordinates you need: a simple pendulum swings one way, so one angle suffices instead of two positions plus a string rule.

Kinetic energy T is half the job: turn half m v squared into your coordinates and rates. For a pendulum length L, T is one half m L squared times rate squared. Potential V is the other half: gravity qualifies, so V is minus m g L cos theta up to a constant.

**Example.** For the pendulum you form L equals T minus V with the single angle theta. That one line carries the whole system: swing rate inside T and height inside V. Friction has no potential and needs separate treatment outside the basic recipe, which is why textbook systems stick to gravity and ideal springs.

Constraint forces like string tension never appear once the coordinates respect the constraint. The angle cannot move against the string, so tension does no work in this description. That disappearance is the payoff for choosing good coordinates.

**Recap.** Pick coordinates the constraints allow, write T minus V, and let tension vanish.

## Practice

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

## Needs first

- [Angular Momentum and Its Conservation](https://lightmysky.com/learn/science/angular-momentum-and-its-conservation-mt_9v3S9-rhwc)
- [Potential Energy Curves, Turning Points and Stability](https://lightmysky.com/learn/science/potential-energy-curves-turning-points-and-stability-mt_ZgRbrDaQzh)

## Opens up

- [Hamilton's Principle and the Action](https://lightmysky.com/learn/science/hamiltons-principle-and-the-action-mt_a273REeeGu)
- [The Euler-Lagrange Equation and Conserved Momenta](https://lightmysky.com/learn/science/the-euler-lagrange-equation-and-conserved-momenta-mt_GSEFABbEXI)