Generalized Coordinates

Generalized coordinates are useful when calculating Lagrangians or Hamiltonians.

The term itself, refers to the parameters that describe the configuration of a system.

For instance, to describe the position of a point on a circle, one could use its rectangular (Cartesian) coordinates, spherical coordinates, or cylindrical coordinates. Buth with generalized coordinates a reference doesn’t need to be made to the specific coordinate system, rather it could be abstracted into something more general as coordinates $(q_{1}, q_{2},…q_{n})$. Describing the position of a point particle in Euclidean space could be described in terms of its Cartesian coordinates, and translating that to generalized coordinates would also require 3 constituents, i.e. $(q_{1}, q_{2}, q_{3})$.

In other words a generalized coordinate system will require $N$ coordinates. So, if there is a system with $N$ particles there would need to be $3N$ coordinates.