Structure and Interpretation of Classical Mechanics

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An analogous result holds when the ‘s depend explicitly on time.

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**b**. Show that, by adding a total time derivative, the Lagrangian can be written in the form , where is a homogeneous quadratic form in the generalized velocities and is independent of velocity.

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The addition of a total time derivative to a Lagrangian does not affect whether the action is critical for a given path.

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Moreover, the additional terms introduced into the action by the total time derivative appear only in the endpoint condition and thus do not affect the Lagrange equations derived from the variation of the action, …

— 1.6.4 The Lagrangian Is Not Unique

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where

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Since is a total time derivative,

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Then, by choosing a suitable function , we can access a simple case that

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This answer is not totally correct, since the generalized velocity, , should be a vector.

— Me@2022-11-01 08:58:52 AM

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2022.11.02 Wednesday (c) All rights reserved by ACHK