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# The Hamiltonian X is always given by g-C = pq L (in one

ISBN: 9781891389221 90

## Solution for problem 13.4 Chapter 13

Classical Mechanics | 0th Edition

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Problem 13.4

The Hamiltonian X is always given by g-C = pq L (in one dimension), and this is the form you should use if in doubt. However, if your generalized coordinate q is "natural" (relation between q and the underlying Cartesian coordinates is independent of time) then J-C = T U, and this form is almost always easier to write down. Therefore, in solving any problem you should quickly check to see if the generalized coordinate is "natural," and if it is you can use the simpler form J-C = T + U. For the Atwood machine of Example 13.2 (page 527), check that the generalized coordinate was "natural." [Hint: There are one generalized coordinate x and two underlying Cartesian coordinates x and y. You have only to write equations for the two Cartesians in terms of the one generalized coordinate and check that they don't involve the time, so it's safe to use g-C = T U. This is ridiculously easy!]

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