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MathidiaMathematics Academy
Analytical mechanics
Lesson 56 of 75
Introduction

Generalised coordinates

Objective. Choose coordinates adapted to constraints.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Generalised coordinates are any convenient set of independent variables that fully describe a mechanical system's configuration, chosen to match its constraints rather than forcing Cartesian axes. For a pendulum of fixed length, the single angle θ fully fixes position, avoiding the redundant constraint x²+y²=l² that Cartesian coordinates would require. The number of generalised coordinates equals the system's degrees of freedom, and choosing them wisely (angles for rotations, arc length for constrained paths) makes equations of motion far simpler. Example: a bead on a circular wire needs only the angle θ, not x and y, since the radius is fixed. Choosing coordinates well is the key skill before applying Lagrangian mechanics.

Worked example

A particle slides on a fixed circle of radius 2. State a suitable generalised coordinate and express its Cartesian position in terms of it.

Diagram of a circle
  1. 1Degrees of freedom: motion is restricted to a circle, so one coordinate suffices
  2. 2Choose generalised coordinate θ, the angle from the positive x-axis
  3. 3Express position: x = 2cosθ, y = 2sinθ
  4. 4This uses exactly one generalised coordinate, consistent with one degree of freedom

Answer: θ, with x = 2cosθ, y = 2sinθ

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Common mistake. Students keep using Cartesian coordinates for constrained systems and then struggle to eliminate constraint forces; choosing an angle or arc length aligned with the constraint removes the need for these forces entirely.

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