---
title: "Degrees of freedom and generalised coordinates"
subject: "Engineering Mechanics"
chapter: 9
key: "semester_3/engineering_mechanics:degrees-of-freedom-generalised-coordinates"
source: https://www.bctengineer.com/q/semester_3/engineering_mechanics/degrees-of-freedom-generalised-coordinates
site: BCT Engineering
---

# Degrees of freedom and generalised coordinates

Engineering Mechanics · Chapter 9: Lagrangian Dynamics

- Asked in: year not given, 5 marks
- Video: [Engineering Dynamics 17.4-01 Degrees of Freedom](https://www.youtube.com/watch?v=MnBoFb1rklw) — Brett Tallman
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/engineering_mechanics/degrees-of-freedom-generalised-coordinates
- Whole chapter: https://www.bctengineer.com/md/semester_3/engineering_mechanics/chapter-9

## Question

Define degrees of freedom and generalised coordinates. Classify constraints as holonomic and non-holonomic. Give the number of degrees of freedom and suitable generalised coordinates for (a) a simple pendulum, (b) a double pendulum, (c) a particle moving in space, (d) a free rigid body in space, and (e) a disc rolling without slipping on a straight line.

## Short answer

DOF = minimum independent coordinates to specify configuration. Generalized coordinates are independent quantities (e.g., angles, displacements) that satisfy constraints. Holonomic constraints are equations among coordinates and time; nonholonomic include non-integrable velocity conditions. Pendulum: 1 DOF, angle θ; double pendulum: 2 DOF, θ₁, θ₂; particle in space: 3 DOF, x,y,z; rigid body: 6 DOF, three translations and three rotations; rolling disc: 1 DOF, position x.

## Answer

**Degrees of freedom (DOF)** is the minimum number of independent coordinates required to completely specify the configuration of a system at any instant. For $N$ particles in space with $c$ independent constraint equations, $n = 3N - c$.

**Generalised coordinates** $q_1, q_2, \dots, q_n$ are any set of $n$ independent quantities (lengths, angles, etc.) that fix the configuration of a system with $n$ degrees of freedom; they need not have the dimension of length, and they automatically satisfy the constraints.

#### Constraints
- **Holonomic:** can be expressed as an equation connecting coordinates (and time) only, $f(x_1, x_2, \dots, t) = 0$; for example a rigid link of fixed length $x^2 + y^2 = l^2$. They reduce the number of coordinates.
- **Non-holonomic:** cannot be integrated into such an equation, being inequalities or non-integrable velocity relations; for example a particle moving on the outside of a sphere (it can leave, $r \ge R$), or a rolling sphere on a rough plane.

Further classes: scleronomic (not explicitly time dependent) and rheonomic (time dependent).

#### Examples

| System | DOF | Generalised coordinates |
|---|---|---|
| (a) Simple pendulum | 1 | angle $\theta$ |
| (b) Double pendulum | 2 | angles $\theta_1,\ \theta_2$ |
| (c) Particle in space | 3 | $x, y, z$ (or $r, \theta, \phi$) |
| (d) Free rigid body in space | 6 | 3 translations of a point + 3 rotation angles (Euler angles) |
| (e) Disc rolling without slip on a line | 1 | $x$ of the centre (angle $\theta = x/r$ follows) |

For the pendulum in (a) the particle has 2 coordinates in the plane $(x, y)$ and one constraint $x^2 + y^2 = l^2$, so $2 - 1 = 1$ DOF. Using generalised coordinates removes the constraint reaction (rod tension) from the equations of motion.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is meant by degrees of freedom?
   A. The number of particles in the system
   B. The minimum number of independent coordinates needed to specify the configuration
   C. The number of forces acting
   D. The number of constraints
   Answer: B. For N particles with c constraints, n = 3N - c. (Key point: Degrees of freedom)

2. Which is a holonomic constraint?
   A. A particle on the outside of a sphere, r >= R
   B. A rigid link, x^2 + y^2 = l^2
   C. A gas molecule in a container
   D. A sphere rolling on a rough plane
   Answer: B. It is an equation in coordinates only; inequalities and non-integrable velocity relations are non-holonomic. (Key point: Holonomic constraints)

3. Which list gives the correct DOF for (a) simple pendulum, (b) double pendulum, (c) particle in space, (d) free rigid body?
   A. 1, 2, 3, 6
   B. 2, 4, 3, 6
   C. 1, 2, 3, 3
   D. 1, 4, 3, 6
   Answer: A. A plane pendulum has 2 coordinates less 1 constraint; the double pendulum has two angles; a rigid body has 3 translations and 3 rotations. (Key point: DOF count)

4. A disc rolling without slipping on a straight line has how many DOF?
   A. 1, since theta = x/r follows from x
   B. 2, x and theta
   C. 3
   D. 0
   Answer: A. The no-slip condition links rotation to translation, leaving the centre position x. (Key point: Rolling disc)

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