36 lines
1.4 KiB
Markdown
36 lines
1.4 KiB
Markdown
---
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title: 13. The reduced mass
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---
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A complete general solution can be obtained for an extremely important
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problem, that of the motion of a system consisting of two interacting particles
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(the two-body problem).
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As a first step towards the solution of this problem, we shall show how it
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can be considerably simplified by separating the motion of the system into
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the motion of the centre of mass and that of the particles relative to the centre
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of mass.
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The potential energy of the interaction of two particles depends only on
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the distance between them, i.e. on the magnitude of the difference in their
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radius vectors. The Lagrangian of such a system is therefore
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L =
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(13.1)
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Let r III r1-r2 - be the relative position vector, and let the origin be at the
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centre of mass, i.e. M1r1+M2r2 = 0. These two equations give
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= m2I/(m1+m2),
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r2
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-M1I/(m1+m2).
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(13.2)
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Substitution in (13.1) gives
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L
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(13.3)
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where
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m=mym
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(13.4)
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is called the reduced mass. The function (13.3) is formally identical with the
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Lagrangian of a particle of mass m moving in an external field U(r) which is
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symmetrical about a fixed origin.
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Thus the problem of the motion of two interacting particles is equivalent
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to that of the motion of one particle in a given external field U(r). From the
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solution r = r(t) of this problem, the paths r1 = r1(t) and r2 = r2(t) of the
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two particles separately, relative to their common centre of mass, are obtained
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by means of formulae (13.2).
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