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Physics / Classical Mechanics 0 / Chapter 7 / Problem 7.45

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

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(a) Verify that the coefficients Ai j in the important

Chapter 7, Problem 7.45

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QUESTION:

(a) Verify that the coefficients Ai j in the important expression (7.94) for the kinetic energy of any "natural" system are symmetric; that is, Ai j = A11. (b) Prove that for any n variables v1, , v dv. . j,k jkViVk [Hint: Start with the case that n = 2, for which you can write out the sums in full. Notice that you need the result of part (a).] This identity is useful in many areas of physics; we needed it to prove the expression (7.96) for the generalized momentum pi.

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QUESTION:

(a) Verify that the coefficients Ai j in the important expression (7.94) for the kinetic energy of any "natural" system are symmetric; that is, Ai j = A11. (b) Prove that for any n variables v1, , v dv. . j,k jkViVk [Hint: Start with the case that n = 2, for which you can write out the sums in full. Notice that you need the result of part (a).] This identity is useful in many areas of physics; we needed it to prove the expression (7.96) for the generalized momentum pi.

ANSWER:

Step 1 of 3

Part (a)

From the expression 7.94:

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