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Variations on the Adiabatic Invariance: the Lorentz Pendulum

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Variations on the Adiabatic Invariance: the Lorentz Pendulum Variations on the adiabatic invariance: the Lorentz pendulum Luis L. Sanchez-Soto´ and Jesus´ Zoido Departamento de Optica,´ Facultad de F´ısica, Universidad Complutense, 28040 Madrid, Spain (Dated: October 17, 2012) We analyze a very simple variant of the Lorentz pendulum, in which the length is varied exponentially, instead of uniformly, as it is assumed in the standard case. We establish quantitative criteria for the condition of adiabatic changes in both pendula and put in evidence their substantially different physical behavior with regard to adiabatic invariance. I. INTRODUCTION concise mathematical viewpoint of Levi-Civitta12,13. The topic of adiabatic invariance has undergone a resur- As early as 1902, Lord Rayleigh1 investigated a pendu- gence of interest from various different fields, such as plasma 14 lum the length of which was being altered uniformly, but physics, thermonuclear research or geophysics , although 15 very slowly (some mathematical aspects of the problem were perhaps Berry’s work on geometric phases has put it again 16 treated in 1895 by Le Cornu2). He showed that if E(t) denotes in the spotlight. The two monographs by Sagdeev et al and 17 the energy and `(t) the length [or, equivalently, the frequency Lochak and Meunier reflect this revival. More recently, the n(t)] at a time t, then issue has renewed its importance in the context of quantum control (for example, concerning adiabatic passage between E(t) E(0) atomic energy levels18–23), as well as adiabatic quantum com- = : (1.1) 24–26 n(t) n(0) putation . Adiabatic invariants are presented in most textbooks in This expression is probably the first explicit example of an terms of action-angle variables, which involves a significant adiabatic invariant; i.e., a conservation law that only holds level of sophistication. Even if a number of pedagogical pa- when the parameters of the system are varied very slowly. The pers has tried to alleviate these difficulties27–34, students often name was coined by analogy with thermodynamics, where understand this notion only at a superficial level. Actually, adiabatic processes are those that occur sufficiently gently. perfunctory application of the adiabaticity condition may lead At the first Solvay Conference in 1911, Lorentz, unaware of to controversial conclusions, even in the hands of experienced Rayleigh’s previous work, raised the question of the behavior practitioners35–38. of a “quantum pendulum” the length of which is gradually al- Quite often the Lorentz pendulum is taken as a typical ex- tered3 (by historical vicissitudes4 his name has become inex- ample to bring up this twist for graduate students39,40. In tricably linked to that system). Einstein’s reply was that “if the spite of its apparent simplicity, the proof of invariance is gen- length of the pendulum is changed infinitely slowly, its energy uinely difficult41–48 and details are omitted. The purpose of remains equal to hn if it is originally hn”, although no detail this paper is to re-elaborate on this topic, putting forth per- of his analysis are given. In the same discussion, Warburg in- tinent physical discussion that emphasizes the motivation for sisted that the length of the pendulum must be altered slowly, doing what is done, as well as to present some variation of the but not systematically. As Arnold aptly remarks5 “the person Lorentz pendulum in the hope that its solution will shed light changing the parameters of the system must no see what state on the subject at an intermediate level. the system is in. Giving this definition a rigorous mathemat- ical meaning is a very delicate and as yet unsolved problem. Fortunately, we can get along with a surrogate. The assump- II. THE UNIFORMLY VARYING PENDULUM tion of ignorance of the internal state of the system on the part of the person controlling the parameter may be replaced by A. Basic equations of motion the requirement that the change of parameter must be smooth; i.e., twice continuously differentiable”. This important point is ignored in most expositions of adiabatic invariance. We confine our attention to the ideal case of a simple pen- Ehrenfest, who did not attend the Solvay Conference dulum of mass m and variable length `(t), oscillating under the gravity. The Lagrangian of the system is arXiv:1210.4241v1 [physics.class-ph] 14 Oct 2012 and was not cognizant of that discussion, had indeed read Rayleigh’s paper and employed those ideas to enunciate his " # 6 1 d`2 dJ 2 famous adiabatic principle , which was promptly reformu- L = m + `2 + mg`cosJ ; (2.1) lated by Born and Fock7 in the form we now call adiabatic 2 dt dt theorem8. In fact, this was a topic of uttermost importance in the old quantum theory9,10. To put it simply, if a physical where J(t) denotes the inclination of the pendulum with the quantity is going to make “all or nothing quantum jumps”, it vertical. The Euler-Lagrange equation for the generalized co- should make no jump at all if the system is perturbed gently, ordinate J becomes and therefore any quantized quantity should be an adiabatic invariant. The reader is referred to the book of Jammer11 for d2J 2 d` dJ g + + sinJ = 0: (2.2) a masterful review of these questions, as well as the lucid and dt2 ` dt dt ` 2 Note in passing that the length `(t) acts as a geometrical (or holonomic) constraint, which here becomes time depen- 0.3 dent49,50. 0.2 In what follows we shall restrict ourselves to the regime of small oscillations (that is, sinJ ' J). In this Section, we 0.1 deal with the example of a pendulum for which the length is uniformly altered in time; i. e., 0.0 `(t) = ` (1 + t); (2.3) 0 e −0.1 where e is a small parameter with the dimensions of a recip- rocal time. It will be convenient to let t = et and define a −0.2 dimensionless time-dependent frequency −0.3 1 r g 0 1 2 3 4 5 w(t) = : (2.4) e `(t) FIG. 1. Exact solution (red continuous line) and asymptotic ap- Equation (2.2) can be thus recast as proximation (blue points) for the uniformly varying pendulum with −1 `˙ e = 0:1 s , `0 = 1 m and J0 = 0:3 rad. J¨ + 2 J˙ + w2(t)J = 0; (2.5) ` where the dot represents differentiation with respect to t. We Consequently, we get limit our analysis to a lengthening pendulum because if the amplitude of the initial displacement is small, then the result- J0 p J(t) ' cos[2w0( 1 + t − 1)]: (2.10) ing displacement will stay small. On the other hand, if the (1 + t)3=4 pendulum is shortening, then even if the initial displacement is small, the subsequent displacement will grow in time, vio- By making use of the approximation (2.10), one can check lating the linearization hypothesis. that the maximum angular amplitude Jmax scales as 51 Using basic properties of Bessel functions , the following 3=4 J0 `0 p p Jmax(t) = = J0 ; (2.11) 1 ( + )3=4 `(t) J(t) = p [AJ1(2w0 1 + t) + BY1(2w0 1 + t)]; 1 t 1 + t (2.6) which shows that it is a decreasing function of time. More- is a solution to (2.5), with w0 = w(0) and Jn(x) and Yn(x) over, limt!¥ Jmax(t) = 0. the Bessel functions of nth order and first and second kind, In Fig. 1 both the exact solution (2.7) and its asymptotic respectively. The constants A and B must be determined by the approximation (2.10) are plotted. The error associated with initial conditions, which we take, without loss of generality, as (2.10) is very small; obviously, this error is smaller when t J(0) = J0 and J˙ (0) = 0. The final result reads as is larger, since larger t’s improve the approximation of (2.8). This can be formally expressed as pJ0w0 p J(t) = p [J2(2w0)Y1(2w0 1 + t) p 1 + t lim 2w0 1 + t = ¥: (2.12) p t!¥ − Y2(2w0)J1(2w0 1 + t)]: (2.7) This limit guarantees that, for a given value of w0, the asymp- As a side comment, we remark that, in spite of its usual des- totic expansion is always a good approximation for any t. ignation, neither Rayleigh nor Lorentz actually examined the From this perspective we can assert that the validity of the behavior of this pendulum; this was first accomplished much 41 asymptotic expansion depends on the values of J0 (and e) but later by Krutkov and Fock , who obtained (2.7) and also de- it is independent of time. rived Eq. (1.1) directly therefrom. Indeed, this solution allows one to investigate in great detail the periods of this system39. Since e appears in the denominator in the definition (2.4), B. Adiabatic invariance w0 is actually very large (for example, if `0 = 1 m and e = −1 0:01 s , then w0 ' 313). This suggests to consider the limit p As pointed out in the Introduction, the concept of adiabatic 2w0 1 + t 1; (2.8) change is associated with a variation that occurs infinitely slowly. The observer who is controlling the changes does not and then take the leading term in the asymptotic expansion of 51 know the internal state of the system. In practical terms, this the Bessel functions means that the change is adiabatic when the variation is car- r 2 p np ried out continuously and so slowly that the change d` of the Jn(x) ∼ cos x − − ; px 4 2 length is very small compared to the length ` of the pendu- (2.9) lum52; i.e.
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