Dynamics

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

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

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7 Hamilton's principle. We can derive at once from the fundamental equation the beautiful theorem known as Hamilton's principle. We consider the motion of the dynamical system in the interval of time from to to t1. Consider now at each instant a virtual displacement (8x1, 8x2, ... , 8x \-) from the position (x1, x2.... , x `,) occupied in the actual motion. The virtual displacement is arbitrary, save that its components 8x1, 8x2, ... , 8xN are functions of t of class C2 which vanish at to and at t1.

2), of the equations of constraint cannot be simplified. The system is non-holonomic. The system has one degree of freedom. In case (iii) a three-fold infinity of positions is accessible from a given position, in case (ii) a two-fold infinity, and in case (i) a single infinity. Case (i) occurs for example if the particle is a bead sliding on a wire of given fixed or varying form. 7) a' bx + b' by + c' bz = 0. 7). 9) 2a + 2'a', Y'=Ab+A'b', Z'=Ac+A'c'. 2). We easily see that the motion is in general determinate.

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. e. 7) where fr = fr(xl, x2, . . , "-"V; t). When this happens the system is said to be holonomic.

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