PlanetPhysics/Generalized Coordinates for Constrained Motion

If the particle is constrained to move on some given surface, any two independent specified 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 $$q_1$$.