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$.



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