PROBLEMS PROBLEM 1. Find a complete integral of the Hamilton-Jacobi equation for motion of a particle in a field U = a/r-Fz (a combination of a uniform field and a Coulomb field). SOLUTION. The field is of the type (48.15), with a(f)=a1F,b(n)a+Fn2 Formula (48.16) gives S = with arbitrary constants Po, E,B. The constant B has in this case the significance that the one- valued function of the co-ordinates and momenta of the particle B is conserved. The expression in the brackets is an integral of the motion for a pure Coulomb field (see \(15). PROBLEM 2. The same as Problem 1, but for a field U = ai/r +az/r2 (the Coulomb field of two fixed points at a distance 2a apart). SOLUTION. This field is of the type (48.21), with a(\)) = (a1+az) /o, = (a1-az)n/o. From formula (48.22) we find S = The constant B here expresses the conservation of the quantity B = cos 01+ cos 02), where M is the total angular momentum of the particle, and 01 and O2 are the angles shown in Fig. 55. 12 r The 20 a FIG. 55 $49. Adiabatic invariants Let us consider a mechanical system executing a finite motion in one dimen- sion and characterised by some parameter A which specifies the properties of the system or of the external field in which it is placed, and let us suppose that 1 varies slowly (adiabatically) with time as the result of some external action; by a “slow” variation we mean one in which A varies only slightly during the period T of the motion: di/dt < A. (49.1) §49 Adiabatic invariants 155 Such a system is not closed, and its energy E is not conserved. However, since A varies only slowly, the rate of change E of the energy is proportional to the rate of change 1 of the parameter. This means that the energy of the system behaves as some function of A when the latter varies. In other words, there is some combination of E and A which remains constant during the motion. This quantity is called an adiabatic invariant. Let H(p, q; A) be the Hamiltonian of the system, which depends on the parameter A. According to formula (40.5), the total time derivative of the energy of the system is dE/dt = OH/dt = (aH/dx)(d)/dt). In averaging this equation over the period of the motion, we need not average the second factor, since A (and therefore i) varies only slowly: dE/dt = (d)/dt) and in the averaged function 01/01 we can regard only P and q, and not A, as variable. That is, the averaging is taken over the motion which would occur if A remained constant. The averaging may be explicitly written dE dt According to Hamilton’s equation q = OHOP, or dt = dq - (CH/OP). The integration with respect to time can therefore be replaced by one with respect to the co-ordinate, with the period T written as here the $ sign denotes an integration over the complete range of variation (“there and back”) of the co-ordinate during the period. Thus dq/(HHap) (49.2) dt $ dq/(HHdp) As has already been mentioned, the integrations in this formula must be taken over the path for a given constant value of A. Along such a path the Hamiltonian has a constant value E, and the momentum is a definite function of the variable co-ordinate q and of the two independent constant parameters E and A. Putting therefore P = p(q; E, 1) and differentiating with respect to A the equation H(p, q; X) )=E, we have = 0, or OH/OP ax ap t If the motion of the system is a rotation, and the co-ordinate q is an angle of rotation , the integration with respect to must be taken over a “complete rotation”, i.e. from 0 to 2nr. 156 The Canonical Equations §49 Substituting this in the numerator of (49.2) and writing the integrand in the denominator as ap/dE, we obtain dt (49.3) dq or dt Finally, this may be written as dI/dt 0, (49.4) where (49.5) the integral being taken over the path for given E and A. This shows that, in the approximation here considered, I remains constant when the parameter A varies, i.e. I is an adiabatic invariant. The quantity I is a function of the energy of the system (and of the para- meter A). The partial derivative with respect to energy is given by 2m DI/DE = $ (ap/dE) dq (i.e. the integral in the denominator in (49.3)) and is, apart from a factor 2n, the period of the motion: (49.6) The integral (49.5) has a geometrical significance in terms of the phase path of the system. In the case considered (one degree of freedom), the phase space reduces to a two-dimensional space (i.e. a plane) with co-ordinates P, q, and the phase path of a system executing a periodic motion is a closed curve in the plane. The integral (49.5) taken round this curve is the area enclosed. It can evidently be written equally well as the line integral I = - $ q dp/2m and as the area integral I = II dp dq/2m. As an example, let us determine the adiabatic invariant for a one-dimen- sional oscillator. The Hamiltonian is H = where w is the frequency of the oscillator. The equation of the phase path is given by the law of conservation of energy H(p, q) = E. The path is an ellipse with semi- axes (2mE) and V(2E/mw2), and its area, divided by 2nr, is I=E/w. (49.7) t It can be shown that, if the function X(t) has no singularities, the difference of I from a constant value is exponentially small. §49 Adiabatic invariants 157 The adiabatic invariance of I signifies that, when the parameters of the oscillator vary slowly, the energy is proportional to the frequency. The equations of motion of a closed system with constant parameters may be reformulated in terms of I. Let us effect a canonical transformation of the variables P and q, taking I as the new “momentum”. The generating function is the abbreviated action So, expressed as a function of q and I. For So is defined for a given energy of the system; in a closed system, I is a func- tion of the energy alone, and so So can equally well be written as a function So(q, I). The partial derivative (So/dq)E is the same as the derivative ( for constant I. Hence (49.8) corresponding to the first of the formulae (45.8) for a canonical trans- formation. The second of these formulae gives the new “co-ordinate”, which we denote by W: W = aso(q,I)/aI. (49.9) The variables I and W are called canonical variables; I is called the action variable and W the angle variable. Since the generating function So(q, I) does not depend explicitly on time, the new Hamiltonian H’ is just H expressed in terms of the new variables. In other words, H’ is the energy E(I), expressed as a function of the action variable. Accordingly, Hamilton’s equations in canonical variables are i = 0, w = dE(I)/dI. (49.10) The first of these shows that I is constant, as it should be; the energy is constant, and I is so too. From the second equation we see that the angle variable is a linear function of time: W = (dE/dI)t + constant. (49.11) The action So(q, I) is a many-valued function of the co-ordinate. During each period this function increases by (49.12) as is evident from the formula So = Spdq and the definition (49.5). During the same time the angle variable therefore increases by Aw = (S/I) = (49.13) t The exactness with which the adiabatic invariant (49.7) is conserved can be determined by establishing the relation between the coefficients C in the asymptotic (t + 00) expressions q = re[c exp(iw+t)] for the solution of the oscillator equation of motion q + w2(t) q = 0. Here the frequency w is a slowly varying function of time, tending to constant limits w as t + 00. The limiting values of I are given in terms of these coefficients by I = tw+/c+l2. The solution is known from quantum mechanics, on account of the formal resemblance between the above equation of motion and SCHRODINGER’S equation 4” + k2(x) 4 = 0 for one-dimensional motion of a particle above a slowly varying (quasi-classical) “potential barrier”. The problem of finding the relation between the asymptotic (x + 00) expressions for & is equivalent to that of finding the “reflection coefficient” of the potential barrier; see Quantum Mechanics, $52, Pergamon Press, Oxford 1965. This method of determining the exactness of conservation of the adiabatic invariant for an oscillator is due to L. P. PITAEVSKII. The relevant calculations are given by A. M. DYKHNE, Soviet Physics JETP 11, 411, 1960. The analysis for the general case of an arbitrary finite motion in one dimension is given by A.A. SLUTSKIN, Soviet Physics JETP 18, 676, 1964. 158 The Canonical Equations