# A Treatise on ANALYTICAL DYNAMICS by L. A. PARS (President of Jesus College, Cambridge)

By L. A. PARS (President of Jesus College, Cambridge)

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4) is clearly satisfied, and we have only to choose f (x) with the properties f(0) = 0, f'(0) = 0; f (x2) = y2, f '(X2) = z2. 7) f (x) = (3y2 - z2x2)(x/x2)2 - (2y2 - z2x2)(x/x2)3 + Ax2(x2 - x)2 + B sin2 (7rx/x2). The number of coordinates less the number of constraints is called the number of degrees of freedom of the system, or, briefly, the number of freedoms. In our present problem, with three coordinates and one constraint, the number of degrees of freedom (or number of freedoms) is two. The important property that we have established is that when the system is holonomic, only a two-fold infinity of positions is accessible from a given starting-point; but when the system is non-holonomic a three-fold infinity of positions is accessible, although there are still only two degrees of freedom.

10) merely serve to determine the multipliers A, A', 2", which are proportional to the forces of constraint, in terms of the given force {X, Y, Z}. 1 A simple example. In the preceding chapter we considered the dynamics of a single particle. It might seem natural, following the historic order of development, to discuss next the theory of the motion of a single rigid body; this is in fact the order usually followed in a first study of rigid Dynamics. Our approach will however be somewhat different.

L < N. DYNAMICAL SYSTEMS 23 The coefficients Ars, Ar are given functions, of class C1, in some domain of (x1, x2, ... , X N; t). These equations are assumed to be independent; they have been reduced to the least possible number. This implies that the matrix of the coefficients has rank L (though in practice the rank may be less than L for isolated values of x, t). 5) r = 1, 2, ... Fs. 6) k = N - L is the number of degrees of freedom of the system; it is the number of velocitycomponents that can be given arbitrary values.