80 lines
4.4 KiB
Text
80 lines
4.4 KiB
Text
PROBLEM 2. Determine the Poisson brackets formed from the components of M.
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SOLUTION. A direct calculation from formula (42.5) gives [Mx, My] = -M2, [My, M]
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= -Mx, [Mz, Mx] = -My.
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Since the momenta and co-ordinates of different particles are mutually independent variables,
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it is easy to see that the formulae derived in Problems 1 and 2 are valid also for the total
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momentum and angular momentum of any system of particles.
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PROBLEM 3. Show that [, M2] = 0, where is any function, spherically symmetrical
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about the origin, of the co-ordinates and momentum of a particle.
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SOLUTION. Such a function can depend on the components of the vectors r and p only
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through the combinations r2, p2, r. p. Hence
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and similarly for The required relation may be verified by direct calculation from
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formula (42.5), using these formulae for the partial derivatives.
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PROBLEM 4. Show that [f, M] = n xf, where f is a vector function of the co-ordinates
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and momentum of a particle, and n is a unit vector parallel to the z-axis.
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SOLUTION. An arbitrary vector f(r,p) may be written as f = where
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01, O2, 03 are scalar functions. The required relation may be verified by direct calculation
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from formulae (42.9), (42.11), (42.12) and the formula of Problem 3.
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$43. The action as a function of the co-ordinates
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In formulating the principle of least action, we have considered the integral
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(43.1)
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taken along a path between two given positions q(1) and q(2) which the system
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occupies at given instants t1 and t2. In varying the action, we compared the
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values of this integral for neighbouring paths with the same values of q(t1)
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and q(t2). Only one of these paths corresponds to the actual motion, namely
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the path for which the integral S has its minimum value.
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Let us now consider another aspect of the concept of action, regarding S
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as a quantity characterising the motion along the actual path, and compare
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the values of S for paths having a common beginning at q(t1) = q(1), but
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passing through different points at time t2. In other words, we consider the
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action integral for the true path as a function of the co-ordinates at the upper
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limit of integration.
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The change in the action from one path to a neighbouring path is given
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(if there is one degree of freedom) by the expression (2.5):
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8S =
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Since the paths of actual motion satisfy Lagrange's equations, the integral
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in 8S is zero. In the first term we put Sq(t1) = 0, and denote the value of
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§43
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The action as a function of the co-ordinates
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139
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8q(t2) by 8q simply. Replacing 0L/dq by p, we have finally 8S = pdq or, in
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the general case of any number of degrees of freedom,
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ES==Pisqu-
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(43.2)
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From this relation it follows that the partial derivatives of the action with
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respect to the co-ordinates are equal to the corresponding momenta:
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=
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(43.3)
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The action may similarly be regarded as an explicit function of time, by
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considering paths starting at a given instant t1 and at a given point q(1), and
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ending at a given point q(2) at various times t2 = t. The partial derivative
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asiat thus obtained may be found by an appropriate variation of the integral.
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It is simpler, however, to use formula (43.3), proceeding as follows.
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From the definition of the action, its total time derivative along the path is
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dS/dt = L.
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(43.4)
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Next, regarding S as a function of co-ordinates and time, in the sense des-
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cribed above, and using formula (43.3), we have
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dS
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A comparison gives asid = L- or
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(43.5)
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Formulae (43.3) and (43.5) may be represented by the expression
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(43.6)
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for the total differential of the action as a function of co-ordinates and time
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at the upper limit of integration in (43.1). Let us now suppose that the co-
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ordinates (and time) at the beginning of the motion, as well as at the end,
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are variable. It is evident that the corresponding change in S will be given
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by the difference of the expressions (43.6) for the beginning and end of the
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path, i.e.
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dsp
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(43.7)
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This relation shows that, whatever the external forces on the system during
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its motion, its final state cannot be an arbitrary function of its initial state;
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only those motions are possible for which the expression on the right-hand
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side of equation (43.7) is a perfect differential. Thus the existence of the
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principle of least action, quite apart from any particular form of the Lagran-
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gian, imposes certain restrictions on the range of possible motions. In parti-
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cular, it is possible to derive a number of general properties, independent
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of the external fields, for beams of particles diverging from given points in
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140
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The Canonical Equations
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