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Unformatted text preview: ation, the amount of energy in the string should
be proportional to
dx 2 L dx =
dx 2 dx + constant. Letting T denote the constant of proportionality yields
L energy in string =
dx 2 dx + constant. Potential energy is only deﬁned up to addition of a constant, so we can drop
the constant term to obtain (4.17).
The force acting on a portion of the string when it is in the conﬁguration
u(x) is determined by an element F (x) of V0 . We imagine that the force acting
on the portion of the string from x to x + dx is F (x)dx. When the force
pushes the string through an inﬁnitesimal displacement ξ (x) ∈ V0 , the total 95 work performed by F (x) is then the “sum” of the forces acting on the tiny
pieces of the string, in other words, the work is the “inner product” of F and ξ ,
L F (x), ξ (x) = F (x)ξ (x)dx.
0 (Note that the inner product we use here diﬀers from the one used in Section 3.3
by a constant factor.)
On the other hand this work is the amount of potential energy lost when the
string undergoes the displacement:
L F (x), ξ (x) =
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This document was uploaded on 01/12/2014.
- Winter '14