88 lines
4.7 KiB
Text
88 lines
4.7 KiB
Text
The Hamilton-Jacobi equation
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147
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in which these repeated quantities are regarded as constant in carrying out
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the differentiations. Hence
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(46.4)
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P=constant
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q=constant
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The Jacobian in the numerator is, by definition, a determinant of order s
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whose element in the ith row and kth column is Representing the
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canonical transformation in terms of the generating function (q, P) as in
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(45.8), we have = In the same way we find that the
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ik-element of the determinant in the denominator of (46.4) is
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This means that the two determinants differ only by the interchange of rows
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and columns; they are therefore equal, so that the ratio (46.4) is equal to
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unity. This completes the proof.
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Let us now suppose that each point in the region of phase space considered
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moves in the course of time in accordance with the equations of motion of the
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mechanical system. The region as a whole therefore moves also, but its volume
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remains unchanged:
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f dr = constant.
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(46.5)
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This result, known as Liouville's theorem, follows at once from the invariance
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of the volume in phase space under canonical transformations and from the
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fact that the change in p and q during the motion may, as we showed at the end
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of §45, be regarded as a canonical transformation.
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In an entirely similar manner the integrals
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11 2 dae dph
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,
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in which the integration is over manifolds of two, four, etc. dimensions in
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phase space, may be shown to be invariant.
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47. The Hamilton-Jacobi equation
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In §43 the action has been considered as a function of co-ordinates and
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time, and it has been shown that the partial derivative with respect to time
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of this function S(q, t) is related to the Hamiltonian by
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and its partial derivatives with respect to the co-ordinates are the momenta.
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Accordingly replacing the momenta P in the Hamiltonian by the derivatives
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as/aq, we have the equation
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(47.1)
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which must be satisfied by the function S(q, t). This first-order partial
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differential equation is called the Hamilton-Jacobi equation.
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148
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The Canonical Equations
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§47
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Like Lagrange's equations and the canonical equations, the Hamilton-
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Jacobi equation is the basis of a general method of integrating the equations
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of motion.
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Before describing this method, we should recall the fact that every first-
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order partial differential equation has a solution depending on an arbitrary
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function; such a solution is called the general integral of the equation. In
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mechanical applications, the general integral of the Hamilton-Jacobi equation
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is less important than a complete integral, which contains as many independent
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arbitrary constants as there are independent variables.
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The independent variables in the Hamilton-Jacobi equation are the time
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and the co-ordinates. For a system with s degrees of freedom, therefore, a
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complete integral of this equation must contain s+1 arbitrary constants.
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Since the function S enters the equation only through its derivatives, one
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of these constants is additive, so that a complete integral of the Hamilton-
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Jacobi equation is
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Sft,q,saas)+
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(47.2)
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where X1, ..., as and A are arbitrary constants.
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Let us now ascertain the relation between a complete integral of the
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Hamilton-Jacobi equation and the solution of the equations of motion which
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is of interest. To do this, we effect a canonical transformation from the
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variables q, P to new variables, taking the function f (t, q; a) as the
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generating function, and the quantities a1, A2, ..., as as the new momenta.
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Let the new co-ordinates be B1, B2, ..., Bs. Since the generating function
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depends on the old co-ordinates and the new momenta, we use formulae
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(45.8): Pi = af/dqi, Bi = af/dar, H' = H+dfdd. But since the function f
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satisfies the Hamilton-Jacobi equation, we see that the new Hamiltonian is
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zero: H' = H+af/dt = H+as/t = 0. Hence the canonical equations in
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the new variables are di = 0, Bi = 0, whence
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ay=constant,
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Bi = constant.
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(47.3)
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By means of the S equations af/dai = Bi, the S co-ordinates q can be expressed
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in terms of the time and the 2s constants a and B. This gives the general
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integral of the equations of motion.
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t Although the general integral of the Hamilton-Jacobi equation is not needed here, we
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may show how it can be found from a complete integral. To do this, we regard A as an arbi-
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trary function of the remaining constants: S = f(t, q1, ..., q8; a1, as) +A(a1, as). Re-
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placing the Ai by functions of co-ordinates and time given by the S conditions asidar = 0,
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we obtain the general integral in terms of the arbitrary function A(a1,..., as). For, when the
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function S is obtained in this manner, we have
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as
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The quantities (as/dqs)a satisfy the Hamilton-Jacobi equation, since the function S(t, q; a)
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is assumed to be a complete integral of that equation. The quantities asida therefore satisfy
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the same equation.
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