Thus once again we need to nd the nontrivial

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Unformatted text preview: Cole Publishing Company, 2000. 94 instant of time. We will assume that the potential energy in the string when it is in the configuration u(x) is L T 2 V (u(x)) = 0 2 du dx dx, (4.17) where T is a constant, called the tension of the string. Indeed, we could imagine that we have devised an experiment that measures the potential energy in the string in various configurations, and has determined that (4.17) does indeed represent the total potential energy in the string. On the other hand, this expression for potential energy is quite plausible for the following reason: We could imagine first that the amount of energy in the string should be proportional to the amount of stretching of the string, or in other words, proportional to the length of the string. From vector calculus, we know that the length of the curve u = u(x) is given by the formula L 1 + (du/dx)2 dx. Length = 0 But when du/dx is small, 1 1+ 2 du dx 22 du dx =1+ 2 + a small error, and hence 1 + (du/dx)2 1 is closely approximated by 1 + (du/dx)2 . 2 Thus to a first order of approxim...
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