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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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Usually we shall assume that each particle has constant mass (though this assumption is not an essential part of the idea of a dynamical system, and later, in Chapter XI, we shall discuss how the more familiar theory is modified if the mass of each particle is a given function of its speed). We denote the coordinates of the particles, referred to fixed rectangular axes, by x1, x2, ... , XN, where N = 3v. The x-, y-, and z-coordinates of the rth particle are x3r_2, x3r_1, x3r. The mass of this particle is denoted indifferently by m3r_2, mar-1, or mar.

But such systems will not be considered in this book. 6 The constrained particle (ii). We now consider a slightly more complicated problem. We suppose that the particle is acted on by a given force as before, but that this time the particle moves, not on a fixed smooth surface, but on a variable smooth surface p(x, y, z, t) = 0, where ' E C2. 3) ay ay axx+ay az av at 0, adx+a-dy+ay dz+av dt=o. 5. For one thing the coefficients contain t as well as x, y, and z; but this is not the vital difference.

The physical meaning of the virtual displacements is that they are displacements with the length of the rod unvaried. It is clear that the problem is in general determinate. 6) m1y1 = YI - A(y2 - y1), m2x2 = X2 + A(x2 - x1), M292 = Y2 + A(y2 - y1), l (x2 - x1)2 + (y2 - 91)2 = a2, suffices to determine the five variables x1, y11 x2, Y21 A as functions of t. 6. Forces of constraint are called into play; the characteristic properties of the forces of constraint are that they do no work, as a whole, in an arbitrary virtual displacement, and that the motion of the system under the action of the two sets of forces together (given forces and forces of constraint) conforms to the equations of constraint.

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