PlanetPhysics/Generalized Coordinates for Constrained Motion

If the [[../Particle/|particle]] is constrained to move on some given surface, any two independent specified [[../Bijective/|functions]] of its rectangular coordinates $x,y,z$, may be taken as its coordinates $q_1$ and $q_2$, provided that by the equation of the given surface in rectangular coordinates and the equations formed by writing $q_1$ and $q_2$ equal to their values in terms of $x,y$, and $z$ the last-named coordinates may be uniquely obtained as explicit functions of $q_1$ and $q_2$.

If the particle is constrained to move in a given path, any specified function of $x,y,z$ may be taken as its coordinate $q_1$, provided that by the two rectangular equations of its path and the equation formed by writing $q_1$ equal to its value in terms of $x,y,z$ the last-named coordinates may be uniquely obtained as explicit functions of $q_1$.