What is meant by degrees of freedom?

The minimum number of independent coordinates needed to specify the configuration

The number of particles in the system

Which is a holonomic constraint?

A particle on the outside of a sphere, r >= R

A sphere rolling on a rough plane

A disc rolling without slipping on a straight line has how many DOF?

1, since theta = x/r follows from x

Practice · 5 marks

Degrees of freedom and generalised coordinates

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.

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.

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.

✓ Answer checked by Claude Opus 5.5 (high effort)

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