By L. A. PARS (President of Jesus College, Cambridge)
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Extra info for A Treatise on ANALYTICAL DYNAMICS
MOTION OF A PARTICLE 17 What is the essential difference between the character of holonomic and of nonholonomic systems? To answer this question it will suffice to consider catastatic systems, and indeed it will suffice to consider the simple case in which the coefficients a, b, c do not depend on t. 2) cp(x, y, z) = constant. 3) Ox, y, z) = 00, 0, 0). On the other hand, if the system is non-holonomic a three fold infinity of positions is accessible. 4) dy - z dx = 0, which clearly does not admit an integrating factor.
Geometrically this result implies that the rate of sweeping out of area by the radius vector is constant, which is the theorem of areas. 13) holds is that of a particle sliding under gravity on a smooth surface of revolution which has a vertical axis Oz; the most familiar case is the problem of the spherical pendulum where the surface is a sphere with its centre on Oz. 3 The catastatic system and the first form of the equation of energy. e. 4) are all identically zero-the class of virtual velocities coincides with the class of possible velocities.
5) together determine the N + L variables x1, x2, ... , xN; Al, A2, ... , AL as functions of t. Theoretically we can determine the motion for some interval containing t = 0 if the values of x's and -Cs at t = 0 are prescribed. The theorem just enunciated expresses the equations of motion for the general dynamical system in a conspicuously simple and comprehensive form. Nevertheless it is not the form that turns out to be the most useful. In practice we are not as a rule interested in the A's (which are related to the magnitudes of the forces of constraint) and (in the next Chapter) we shall transform the equations into a different form in which the A's do not appear.