57 lines
2.4 KiB
Markdown
57 lines
2.4 KiB
Markdown
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title: Determination of the potential energy from the period of oscillation
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Let us consider to what extent the form of the potential energy U(x) of a
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field in which a particle is oscillating can be deduced from a knowledge of the
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period of oscillation T as a function of the energy E. Mathematically, this
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involves the solution of the integral equation (11.5), in which U(x) is regarded
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as unknown and T(E) as known.
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We shall assume that the required function U(x) has only one minimum
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in the region of space considered, leaving aside the question whether there
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exist solutions of the integral equation which do not meet this condition.
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For convenience, we take the origin at the position of minimum potential
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energy, and take this minimum energy to be zero (Fig. 7).
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U
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U=E
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121U
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X
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X1
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X2
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FIG. 7
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28
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Integration of the Equations of Motion
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§12
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In the integral (11.5) we regard the co-ordinate x as a function of U. The
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function x(U) is two-valued: each value of the potential energy corresponds
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to two different values of X. Accordingly, the integral (11.5) must be divided
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into two parts before replacing dx by (dx/dU) dU: one from x = X1 to x = 0
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and the other from x = 0 to X = X2. We shall write the function x(U) in
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these two ranges as x = x1(U) and x = x2(U) respectively.
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The limits of integration with respect to U are evidently E and 0, so that
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we have
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T(E) =
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=
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If both sides of this equation are divided by V(a - E), where a is a parameter,
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and integrated with respect to E from 0 to a, the result is
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dUdE
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or, changing the order of integration,
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dE
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The integral over E is elementary; its value is TT. The integral over U is
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thus trivial, and we have
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since x2(0) = x1(0) = 0. Writing U in place of a, we obtain the final result:
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(12.1)
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Thus the known function T(E) can be used to determine the difference
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x2(U)-x1(U). - The functions x2(U) and x1(U) themselves remain indeter-
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minate. This means that there is not one but an infinity of curves U = U(x)
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§13
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The reduced mass
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29
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which give the prescribed dependence of period on energy, and differ in such
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a way that the difference between the two values of x corresponding to each
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value of U is the same for every curve.
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The indeterminacy of the solution is removed if we impose the condition
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that the curve U = U(x) must be symmetrical about the U-axis, i.e. that
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x2(U) = 1(U) III x(U). In this case, formula (12.1) gives for x(U) the
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unique expression
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(12.2)
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