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Unformatted text preview: th a large or even
inﬁnite number of unknowns can give an eﬀective model for the same physical
phenomenon. This partially explains, for example, why quantum mechanics
possesses two superﬁcially diﬀerent formulations, via Schr¨dinger’s partial difo
ferential equation or via “inﬁnite matrices” in Heisenberg’s “matrix mechanics.” 4.4 The vibrating string Our next goal is to derive the equation which governs the motion of a vibrating
string. We consider a string of length L stretched out along the x-axis, one end
of the string being at x = 0 and the other being at x = L. We assume that the
string is free to move only in the vertical direction. Let
u(x, t) = vertical displacement of the string at the point x at time t.
We will derive a partial diﬀerential equation for u(x, t). Note that since the
ends of the string are ﬁxed, we must have u(0, t) = 0 = u(L, t) for all t.
It will be convenient to use the “conﬁguration space” V0 described in Section 3.3. An element u(x) ∈ V0 represents a conﬁguration of the string at some
3 For further discussion of this method one can refer to numerical analysis books, such as
Burden and Faires, Numerical analysis , Seventh edition, Brooks...
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This document was uploaded on 01/12/2014.
- Winter '14