Past exam of the physics 1 paper of the University of Oxford / 2014 / 4

Generalized coordinates are variables $q_1,\dots ,q_n$ that specify the allowed instantaneous configuration of a mechanical system. They need not be Cartesian positions: a rigid pendulum's angle is a suitable single coordinate. With constraints solved, the particle positions are functions ${\mathbf{r}}_i(q_1,\dots ,q_n,t)$.

The number of degrees of freedom is the number of independent coordinates needed to specify an allowed configuration locally. For $N$ unconstrained point particles in three dimensions it is $3N$; $k$ independent regular holonomic constraints reduce it to $3N-k$. Redundant constraints must not be counted twice, and nonholonomic velocity restrictions require separate care.

A holonomic constraint can be written as an equation $f(q_1,\dots ,q_n,t)=0$ involving coordinates and possibly time, without an irreducible dependence on velocities. For example a particle confined to a fixed sphere obeys $x^2+y^2+z^2-a^2=0$. A general rolling without slipping constraint in three dimensions need not be integrable into such a coordinate equation.