Generalized coordinates and configuration space

For N point particles in three dimensions, unconstrained configurations require 3N local coordinates. Suppose k smooth position constraints have independent gradients near the arrangement of interest. Their common zero set then has local dimension 3N−k. Independence is essential: writing the same equation twice removes only one freedom. A configuration-space point specifies positions, not velocities. A moving system traces a curve in this space, which need not resemble any one particle's path in the laboratory.

A plane pendulum illustrates the distinction. Its bob occupies a circle, so one angle labels local configurations even though the drawing uses two axes. The angle is periodic: zero and a full turn represent the same arrangement. A single real interval therefore cannot be a globally unique coordinate chart for the whole circle. This is a coordinate issue, not an extra mechanical force. Before differentiating, state the coordinate range, the reference direction, and any configurations where the chosen chart fails.

Write every position as a function of independent coordinates and time. The chain rule turns coordinate rates into physical velocities. For a stationary constraint, velocity is a sum of tangent vectors weighted by coordinate rates. A moving track adds an explicit time derivative even when every generalized coordinate is held fixed. Omitting this term silently removes the motion imposed by the apparatus. Kinetic energy must be computed from laboratory velocities before deciding which terms can be neglected.

For stationary constraints and fixed positive masses, kinetic energy is a quadratic form in the rates. Its matrix is the mass-weighted inner product of the coordinate tangent vectors. It is positive definite when those vectors are independent. Off-diagonal entries describe geometric coupling; they do not imply an additional spring or interaction. In polar coordinates, the angular entry contains radius squared because the same angular rate corresponds to a greater physical speed farther from the origin.

A free particle in a plane has two configuration coordinates. Changing from Cartesian to polar coordinates does not constrain it. At the origin, however, the angular tangent vector vanishes, so the polar mass matrix loses rank even though Cartesian motion remains well defined. This is why a singular matrix can indicate a bad chart rather than a new physical law. Return to a nonsingular coordinate system and check the original position map before declaring the mechanical model defective.

The vector lab below isolates one ingredient of this construction: how an angle and magnitude determine Cartesian projections. Predict the signs at an obtuse angle, then compare its displayed components with your position map. The magnitude control is a vector length, not a dynamically evolved radial coordinate. Playback does not solve the pendulum or infer a configuration-space metric. Use the picture to check geometry, and use differentiation and kinetic energy to establish the mechanical model.