Analytical mechanics

Generalized coordinates and degrees of freedom

Generalized coordinates describe a system's configuration after constraints are accounted for; the number of independent coordinates is its number of degrees of freedom.

A simple pendulum does not need three independent Cartesian coordinates: the inextensible string and fixed pivot restrict the bob to an arc. Its angle $\theta$ suffices. A suitable coordinate builds the constraint into the description.

Holonomic constraints

A holonomic constraint is an equation among coordinates, possibly involving time; for a planar pendulum, $x^2+y^2=l^2$. One equation constrains two Cartesian coordinates, leaving one independent coordinate. A velocity constraint that cannot generally be integrated is nonholonomic.

Replacing $x,y$ by one angle automatically enforces the fixed string length. In many-body systems, generalized coordinates may also be joint angles, distances, or collective variables; they need not be geometric lengths measured in metres.

Quick check

View the allowed configurations as points on a configuration manifold. If holonomic constraints are $f_a(\mathbf x,t)=0$, the Jacobian $\partial f_a/\partial x_i$ identifies independent constraints locally; its rank determines the manifold’s local dimension. For a time-dependent constraint, the allowed configuration set itself moves, so admissible velocities must obey $\sum_i(\partial f_a/\partial x_i)\dot x_i+\partial f_a/\partial t=0$. Angles, arc lengths, and Cartesian coordinates are all valid choices when they specify the configuration independently.

Two free particles in space have six coordinates; fixing their separation imposes one independent constraint, leaving five degrees of freedom. Choosing center-of-mass coordinates together with suitable angular and relative coordinates can make that constraint explicit.