Show that in the cylindrical (polar) coordinates (r, 6, z), which are related to the rectangular coor- dinates by x = r cos o, y = r sin o, z = z, the z-component of the angular momentum of a particle with mass m, M, = m(xỷ – yà), is equal to M, = mr²6. Then show that M2 = Pø = те where L = (1/2)m(i2 + r²6² + ¿²) – U(r, ø, z) is the Lagrangian for the particle in the cylindrical coordinates. This exercise shows that the angular momentum is a generalized momentum corre- sponding to an angle regarded as a generalized coordinate.
Show that in the cylindrical (polar) coordinates (r, 6, z), which are related to the rectangular coor- dinates by x = r cos o, y = r sin o, z = z, the z-component of the angular momentum of a particle with mass m, M, = m(xỷ – yà), is equal to M, = mr²6. Then show that M2 = Pø = те where L = (1/2)m(i2 + r²6² + ¿²) – U(r, ø, z) is the Lagrangian for the particle in the cylindrical coordinates. This exercise shows that the angular momentum is a generalized momentum corre- sponding to an angle regarded as a generalized coordinate.
Classical Dynamics of Particles and Systems
5th Edition
ISBN:9780534408961
Author:Stephen T. Thornton, Jerry B. Marion
Publisher:Stephen T. Thornton, Jerry B. Marion
Chapter10: Motion In A Noninertial Reference Frame
Section: Chapter Questions
Problem 10.12P
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