103 lines
4.5 KiB
Text
103 lines
4.5 KiB
Text
space. The study of these properties forms a part of the subject of geometrical
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optics.+
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It is of interest to note that Hamilton's equations can be formally derived
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from the condition of minimum action in the form
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(43.8)
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which follows from (43.6), if the co-ordinates and momenta are varied inde-
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pendently. Again assuming for simplicity that there is only one co-ordinate
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and momentum, we write the variation of the action as
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= dt - (OH/dp)8p dt].
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An integration by parts in the second term gives
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At the limits of integration we must put 8q = 0, so that the integrated term
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is zero. The remaining expression can be zero only if the two integrands
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vanish separately, since the variations Sp and 8q are independent and arbitrary
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dq = (OH/OP) dt, dp = - (dH/dq) dt, which, after division by dt, are
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Hamilton's equations.
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$44. Maupertuis' principle
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The motion of a mechanical system is entirely determined by the principle
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of least action: by solving the equations of motion which follow from that
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principle, we can find both the form of the path and the position on the path
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as a function of time.
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If the problem is the more restricted one of determining only the path,
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without reference to time, a simplified form of the principle of least action
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may be used. We assume that the Lagrangian, and therefore the Hamilton-
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ian, do not involve the time explicitly, SO that the energy of the system is
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conserved: H(p, q) = E = constant. According to the principle of least action,
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the variation of the action, for given initial and final co-ordinates and times
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(to and t, say), is zero. If, however, we allow a variation of the final time t,
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the initial and final co-ordinates remaining fixed, we have (cf.(43.7))
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8S = -Hot.
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(44.1)
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We now compare, not all virtual motions of the system, but only those
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which satisfy the law of conservation of energy. For such paths we can
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replace H in (44.1) by a constant E, which gives
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SS+Est=0.
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(44.2)
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t See The Classical Theory of Fields, Chapter 7, Pergamon Press, Oxford 1962.
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§44
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Maupertuis' principle
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141
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Writing the action in the form (43.8) and again replacing H by E, we have
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(44.3)
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The first term in this expression,
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(44.4)
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is sometimes called the abbreviated action.
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Substituting (44.3) in (44.2), we find that
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8S0=0.
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(44.5)
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Thus the abbreviated action has a minimum with respect to all paths which
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satisfy the law of conservation of energy and pass through the final point
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at any instant. In order to use such a variational principle, the momenta
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(and so the whole integrand in (44.4)) must be expressed in terms of the
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co-ordinates q and their differentials dq. To do this, we use the definition of
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momentum:
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(44.6)
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and the law of conservation of energy:
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E(g)
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(44.7)
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Expressing the differential dt in terms of the co-ordinates q and their differen-
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tials dq by means of (44.7) and substituting in (44.6), we have the momenta
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in terms of q and dq, with the energy E as a parameter. The variational prin-
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ciple so obtained determines the path of the system, and is usually called
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Maupertuis' principle, although its precise formulation is due to EULER and
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LAGRANGE.
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The above calculations may be carried out explicitly when the Lagrangian
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takes its usual form (5.5) as the difference of the kinetic and potential energies:
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The momenta are
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and the energy is
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The last equation gives
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dt
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(44.8)
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142
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The Canonical Equations
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§44
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substituting this in
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Epides
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we find the abbreviated action:
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(44.9)
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In particular, for a single particle the kinetic energy is T = 1/2 m(dl/dt)2,
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where m is the mass of the particle and dl an element of its path; the variational
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principle which determines the path is
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${/[2m(B-U)]dl=0
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(44.10)
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where the integral is taken between two given points in space. This form is
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due to JACOBI.
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In free motion of the particle, U = 0, and (44.10) gives the trivial result
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8 I dl = 0, i.e. the particle moves along the shortest path between the two
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given points, i.e. in a straight line.
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Let us return now to the expression (44.3) for the action and vary it with
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respect to the parameter E. We have
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substituting in (44.2), we obtain
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(44.11)
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When the abbreviated action has the form (44.9), this gives
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=
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(44.12)
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which is just the integral of equation (44.8). Together with the equation of
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the path, it entirely determines the motion.
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PROBLEM
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Derive the differential equation of the path from the variational principle (44.10).
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SOLUTION. Effecting the variation, we have
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f
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In the second term we have used the fact that dl2 = dr2 and therefore dl d8l = dr. d&r.
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Integrating this term by parts and then equating to zero the coefficient of Sr in the integrand,
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we obtain the differential equation of the path:
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