Degrees of freedom

Remarks

Example: $N$ point particles in $\R^3$ with $k$ independent holonomic constraints have $3N - k$ degrees of freedom. A planar double pendulum has $2$ particles in the plane, so $4$ coordinates, and $2$ rod constraints; it has $4 - 2 = 2$ degrees of freedom. Non-example: a set of positions that is not a manifold, such as two crossing lines, has no well-defined dimension near the crossing. There the count of degrees of freedom is not defined. If the system also has $m$ independent nonholonomic constraints on the velocities, it has $n - m$ independent velocities. Some authors call $n - m$ the number of degrees of freedom and keep $n$ as the number of generalized coordinates.