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 q1 and q2, provided
that by the equation of the given surface in rectangular coordinates and the equations formed by
writing q1 and q2 equal to their values in terms of x,y,z, the last-named coordinates may be
uniquely obtained as explicit functions of q1 and q2. 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 q1, provided that by the two rectangular equations of its path and the equation
formed by writing q1 equal to its value in terms of x,y,z, the last-named coordinates may be
uniquely obtained as explicit functions of q1. 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.
