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

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

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9) we see that X1', X2', ... 12) msxs = XS + XS', s = 1, 2, ... s, s = 1, 2, ... , N. 5) 1 Arsxs +Ar=0, r = 1, 2,... 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.

1(6T Jto + I X 8x) dt = -J 1{_I (mx - X) ox} dt. 4) + I X bx) dt = 0. f to This is Hamilton's principle. 4) in the form to 8(T - V) dt = 0. J f to to In applying the principle we must remember that the terminal points are fixed, and so are the instants of departure and arrival. e. it is a necessary condition for the motion. In fact the condition is also sufficient. e. 4) holds for an arbitrary variation of the type prescribed, then the original motion is the actual (dynamically possible) motion of the system.

2) is the first, or primitive, form of the equation of energy. It expresses the fact that the rate of increase of the kinetic energy is equal to the rate of working of the given forces. The equation holds for any catastatic system, even for systems for which the coefficients Ars in the equations of constraint depend upon t. 4 Conservative forces and the second form of the equation of energy. Hitherto we have thought of the x's as the actual coordinates at time t in some particular motion of the system.

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


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