Dynamics of a Particle: 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 $x,y,z$ , the last-named coordinates may be uniquely obtained as explicit functions of $q_1$ and $q_2$ . For when this is done, the reasoning of Art. 2 will hold good.

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 $q_1$ . For when this is done, the reasoning of Art. 2 will hold good.

Source

William Elwood Byerly, An Introduction to the Use of Generalized Coördinates in mechanics and Physics, Ginn and Company, 1916. Chapter I, “Introduction.”

The 1916 source work is in the public domain in the United States.

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