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# Statement of a problem № 3450

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Briefly explain what is the "generalized (or canonical) momentum conjugate to a generalized coordinate". What characteristic feature should the Lagrangian function have for a generalized momentum to be a constant of motion? A particle P can slide on a frictionless horizontal table with a small opening at O. It is attached, by a string of length l passing through the open- ing, to a particle Q hanging vertically under the table (see Fig. 7.9). The two particles have equal mass, m. Let t denote the distance of P to the opening, T the angle between OP and some fixed line through O and g the acceleration of gravity. Initially, r = a, Q does not move, and P is given an initial velocity of magnitude (ag)^1/2 at right angles to OP. (a) Write the Lagrangian in terms of the coordinates r and T and derive the corresponding equations of motion. (b) Using these equations of motion and the initial

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