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Physics / Classical Mechanics 0 / Chapter 9 / Problem 9.11

Classical Mechanics | 0th Edition | ISBN: 9781891389221 | Authors: John R Taylor

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In this problem you will prove the equation of motion

Chapter 9, Problem 9.11

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QUESTION:

In this problem you will prove the equation of motion (9.34) for a rotating frame using the Lagrangian approach. As usual, the Lagrangian method is in many ways easier than the Newtonian (except that it calls for some slightly tricky vector gymnastics), but is perhaps less insightful. Let S be a noninertial frame rotating with constant angular velocity St relative to the inertial frame So. Let both frames have the same origin, 0 = 0'. (a) Find the Lagrangian L = T U in terms of the coordinates r and r of S. [Remember that you must first evaluate T in the inertial frame. In this connection, recall that vo = v + St x r.] (b) Show that the three Lagrange equations reproduce (9.34) precisely.

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QUESTION:

In this problem you will prove the equation of motion (9.34) for a rotating frame using the Lagrangian approach. As usual, the Lagrangian method is in many ways easier than the Newtonian (except that it calls for some slightly tricky vector gymnastics), but is perhaps less insightful. Let S be a noninertial frame rotating with constant angular velocity St relative to the inertial frame So. Let both frames have the same origin, 0 = 0'. (a) Find the Lagrangian L = T U in terms of the coordinates r and r of S. [Remember that you must first evaluate T in the inertial frame. In this connection, recall that vo = v + St x r.] (b) Show that the three Lagrange equations reproduce (9.34) precisely.

ANSWER:

Step 1 of 6

The expression for Lagrangian as

Here, is the kinetic energy and is the potential energy.

The Lagrangian equation:

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