Generalized coordinates

Generalized coordinates are independent parameters that specify a mechanical system’s configuration. They can be chosen to represent constraints directly, reducing the number of variables needed to describe motion.

Configuration space: A space whose points represent the possible configurations of a system. A generalized coordinate tuple identifies a point in this space.
Cartesian coordinates: Coordinates that locate points using perpendicular axes, such as x, y, and z. Generalized coordinates need not be Cartesian or tied to a fixed set of axes.
Simple pendulum: An ideal point mass suspended from a fixed pivot by a massless, inextensible rod. One angle describes its motion while automatically enforcing fixed rod length.
Redundant coordinates: A description using more variables than the system has independent degrees of freedom. They retain dependent coordinates instead of reducing the description to independent parameters.
Degrees of freedom: The number of independent parameters needed to specify a system’s configuration. This count determines how many generalized coordinates are required.
Polar coordinates: A coordinate system that locates planar points by radius and angle. Radius and angle can directly describe a constrained planar mechanism.
Double pendulum: A pair of pendulums connected in series, with motion governed by coupled nonlinear equations. Two joint angles describe its configuration more directly than Cartesian positions.
Generalized velocity: A rate of change associated with a generalized coordinate, often denoted by its time derivative. Velocities describe motion, whereas generalized coordinates specify configuration.
Holonomic constraint: A constraint expressible as an equation among coordinates and possibly time. Such constraints reduce the independent coordinates needed to describe a system.
Virtual displacement: An infinitesimal change in configuration consistent with a system’s constraints at a fixed time. Its components are variations of the generalized coordinates.

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