Generalized Coordinates and Constraints

Accordingly, the position of the mass is completely determined by specifying the angle i , which may assume any value and is hence not subject to a constraint. This angular coordinate is much better adapted to the symmetry of the system since it automatically respects the constraint. Unconstrained coordinates of this type are called generalized coordinates. This new coordinate i and the Cartesian coordinates x and y are uniquely related to one another according to 

Any mechanical system of N particles in R, which is subject to R constraints, features in general / = 3N — R degrees of freedom represented by generalized coordinates i, 2/ / 7/ compatible with all constraints. These generalized coordinates may assume any value independently of each other, and the position of all N particles is uniquely determined by the / generalized coordinates. 

For a given physical system the choice of generalized coordinates is not unique, and often there exists another set of generalized coordinates Q, (i = 1.which is equally well suited for the description of the system. It is a matter of personal taste and experience which set to choose. Obviously, generalized coordinates do not necessarily have to feature the dimension of length. 

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