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Find the Lagrangian function

Find the Lagrangian function. Please see the attachment. The bottom part of the attachment that got cut off just says "of motion of the particle."

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The solution is attached below in three files that are identical in content but differ in file format.
One is in Adobe acrobat (pdf)
One is in MS-Word 2000 (doc)
and one in MS-Word 95 (doc)

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Solution Summary

The solution is explained and given as an equational process in the attachment.