HW2_prob2_GPS19_G17 - Goldstein 1.17(3rd ed 1.19 Because...

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Goldstein 1.17 (3 rd ed. 1.19) Because the mass point is suspended by a rigid rod of length a , the system has only two degrees of freedom. The two generalized coordinates may be taken to be the polar and azimuthmal angles of a spherical coordinate system centered on the point of attachment of the rigid rod, and we will let the polar ( z ) axis point downwards so that the polar angle θ is positive when the mass point is displaced away from its rest position. The kinetic energy of the mass point is T = m 2 ( · x 2 + · y 2 + · z 2 ) . (1) In the spherical coordinate system we have x = a sin θ cos φ , (2) y = a sin θ sin φ , (3) z = a cos θ , (4) from which it follows that · x = a [ · θ cos θ cos φ · φ sin θ sin φ ] , (5) · y = a [ · θ cos θ sin φ + · φ sin θ cos φ ] , (6) · z = a · θ sin θ . (7) After substituting Eqs.(5)—(7) into Eq.(1) we obtain T = 1 2 ma 2 [ · θ 2 + · φ 2 sin 2 θ ] . (8) Since the z axis points down in our coordinmate system, the potential energy is V = mgz = mga cos θ . (9) Thus, the Lagrangian of the pendulum is
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