37 lines
2.1 KiB
Text
37 lines
2.1 KiB
Text
variables at time t, and qt+r, Pt+r their values at another time t +T. The latter
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are some functions of the former (and involve T as a parameter):
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If these formulae are regarded as a transformation from the variables Qt, Pt
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to qt+r, Pttr, then this transformation is canonical. This is evident from the
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expression ds = for the differential of the action S(qt++,
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qt) taken along the true path, passing through the points qt and qt++ at given
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times t and t + T (cf. (43.7)). A comparison of this formula with (45.6) shows
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that - S is the generating function of the transformation.
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46. Liouville's theorem
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For the geometrical interpretation of mechanical phenomena, use is often
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made of phase space. This is a space of 2s dimensions, whose co-ordinate axes
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correspond to the S generalised co-ordinates and S momenta of the system
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concerned. Each point in phase space corresponds to a definite state of the
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system. When the system moves, the point representing it describes a curve
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called the phase path.
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The product of differentials dT = dq1 ... dqsdp1 dps may be regarded
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as an element of volume in phase space. Let us now consider the integral
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I dT taken over some region of phase space, and representing the volume of
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that region. We shall show that this integral is invariant with respect to
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canonical transformations; that is, if the variables P, q are replaced by
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P, Q by a canonical transformation, then the volumes of the corresponding
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regions of the spaces of P, and P, Q are equal:
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...dqsdp1...dps =
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(46.1)
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The transformation of variables in a multiple integral is effected by the
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formula I .jdQ1...dQsdP1...dPz = S... I Ddq1 dp1...dps,
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where
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(46.2)
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is the Jacobian of the transformation. The proof of (46.1) therefore amounts
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to proving that the Jacobian of every canonical transformation is unity:
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D=1.
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(46.3)
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We shall use a well-known property of Jacobians whereby they can be
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treated somewhat like fractions. "Dividing numerator and denominator" by
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0(91, ..., qs, P1, Ps), we obtain
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Another property of Jacobians is that, when the same quantities appear in
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both the partial differentials, the Jacobian reduces to one in fewer variables,
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