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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.

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Show that in the cylindrical (polar) coordinates (r, ø, z), which are related to the rectangular coor- dinates by x = r cos ø, y = r sin ø, z = z, the z-component of the angular momentum of a particle with mass m, M, = m(xý – yi), is equal to M, = mr²ó. Then show that Mz = Pø = Te where L = (1/2)m(r2 + r²o² + ¿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.
Transcribed Image Text:Show that in the cylindrical (polar) coordinates (r, ø, z), which are related to the rectangular coor- dinates by x = r cos ø, y = r sin ø, z = z, the z-component of the angular momentum of a particle with mass m, M, = m(xý – yi), is equal to M, = mr²ó. Then show that Mz = Pø = Te where L = (1/2)m(r2 + r²o² + ¿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.
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