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 ν(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 ν(t) ν(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 hν if it is originally hν”, 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 mathematical 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 dϑ 2 famous adiabatic principle , which was promptly reformu- L = m + `2 + mg`cosϑ , (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 ϑ(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 ϑ becomes and therefore any quantized quantity should be an adiabatic invariant. The reader is referred to the book of Jammer11 for d2ϑ 2 d` dϑ g + + sinϑ = 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, sinϑ ' ϑ). 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 ε −0.1 where ε is a small parameter with the dimensions of a recip- rocal time. It will be convenient to let τ = εt and define a −0.2 dimensionless time-dependent frequency −0.3 1 r g 0 1 2 3 4 5 ω(τ) = . (2.4) ε `(τ) 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 `˙ ε = 0.1 s , `0 = 1 m and ϑ0 = 0.3 rad. ϑ¨ + 2 ϑ˙ + ω2(τ)ϑ = 0, (2.5) ` where the dot represents differentiation with respect to τ. We Consequently, we get limit our analysis to a lengthening pendulum because if the amplitude of the initial displacement is small, then the result- ϑ0 √ ϑ(τ) ' cos[2ω0( 1 + τ − 1)]. (2.10) ing displacement will stay small. On the other hand, if the (1 + τ)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 ϑmax scales as 51 Using basic properties of Bessel functions , the following 3/4 ϑ0 `0 √ √ ϑmax(τ) = = ϑ0 , (2.11) 1 ( + )3/4 `(τ) ϑ(τ) = √ [AJ1(2ω0 1 + τ) + BY1(2ω0 1 + τ)], 1 τ 1 + τ (2.6) which shows that it is a decreasing function of time. More- is a solution to (2.5), with ω0 = ω(0) and Jn(x) and Yn(x) over, limτ→∞ ϑmax(τ) = 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 τ ϑ(0) = ϑ0 and ϑ˙ (0) = 0. The final result reads as is larger, since larger τ’s improve the approximation of (2.8). This can be formally expressed as πϑ0ω0 √ ϑ(τ) = √ [J2(2ω0)Y1(2ω0 1 + τ) √ 1 + τ lim 2ω0 1 + τ = ∞. (2.12) √ τ→∞ − Y2(2ω0)J1(2ω0 1 + τ)]. (2.7) This limit guarantees that, for a given value of ω0, the asymp- As a side comment, we remark that, in spite of its usual des- totic expansion is always a good approximation for any τ. 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 ϑ0 (and ε) 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 ε appears in the denominator in the definition (2.4), B. Adiabatic invariance ω0 is actually very large (for example, if `0 = 1 m and ε = −1 0.01 s , then ω0 ' 313). This suggests to consider the limit √ As pointed out in the Introduction, the concept of adiabatic 2ω0 1 + τ 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 π nπ ried out continuously and so slowly that the change δ` of the Jn(x) ∼ cos x − − , πx 4 2 length is very small compared to the length ` of the pendu- (2.9) lum52; i.e. r 2 π nπ δ` Yn(x) ∼ sin x − − . 1. (2.13) πx 4 2 ` 3

By considering that the temporal interval in which the variation δ` is produced coincides with the local period T for the 0.92 length `, we have 0.90 d` δ` = T , (2.14) dt 0.88 p and recalling that T = 2π `/g, we can rewrite (2.13) for the 0.86 example at hand as 0.84 δ` εT 2π = √ 0 = √ 1, (2.15) ` 1 + τ ω0 1 + τ 0.82 with T0 being the period of the pendulum at τ = 0. This re- 0.0 0.5 1.0 1.5 2.0 quirement is independent of ϑ0 and, thus, independent of the amplitude of the oscillations, which is intuitively expected. Equation (2.15) clearly suggests that if the change in length FIG. 2. Plot of I(τ) as a function of τ for the uniformly vary- is initially adiabatic, it will remain forever. Moreover, the adi- ing pendulum with `0 = 1 m and ϑ0 = 0.3 rad. The curves corre- sponds to ω = 2,10 and 1000 (which are associated with the values abatic character of the system will improve as time goes on. 0 ε = 0.2491,0.0498 and 0.00005 s−1), following the decreasing am- Interestingly enough, condition (2.15) is formally equiva- plitudes. lent to (2.8) ensuring the validity of the asymptotic approximation; they will always be satisﬁed whenever which immediately leads to = , ω0 ωlim 2π (2.16) √ √ H(τ) = H π2ω2[H 2 (2ω 1 + τ) + H 2 (2ω 1 + τ)]. which, from the deﬁnition of ε, can be equivalently recast as 0 0 22 0 21 0 (2.22) 1 r g Here, for notational simplicity, we have introduced the func- ε εlim = . (2.17) tions 2π `0 (x) = J (2ω ) Y (x) −Y (2ω ) J (x), Equations (2.16) or (2.17) (which do not depend on time) pro- H21 2 0 1 2 0 1 vide a sensible criterion for the adiabatic change in a uni- (2.23) formly varying pendulum. The lesser the initial length, the H22(x) = Y2(2ω0) J2(x) − J2(2ω0) Y2(x), more quickly can be lengthening the pendulum under the adi- 2 2 2 and H0 = ϑ0ω ε ` /2 is the total energy at τ = 0. Finally, abatic hypothesis. Alternatively, ωlim can be seen as the mini- 0 √ 0 since ( ) = / 1 + , with v = /2 , we get mum value of ω0 for which the asymptotic expansion is valid ν τ ν0 τ 0 ω0ε π independently of τ. H √ I(τ) = 0 1 + τ π2ω2 To give a more quantitative argument, we compute the func- ν 0 0 √ √ tion 2 2 × [H22(2ω0 1 + τ) + H21(2ω0 1 + τ)]. (2.24) H(τ) I(τ) = , (2.18) I( ) ν(τ) Because τ is a time-dependent function, it will not be, in general, an adiabatic invariant. In other words, for arbitrary that turns out to be an adiabatic invariant for arbitrary periodic values of the parameters, the solution (2.7) will not be as- motions in one degree of freedom6. Here H(τ) is the Hamil- sociated with adiabatic changes in the length of the pendu- tonian and ν(τ) the frequency of the oscillations. lum. The function I(τ) is shown in Fig. 2 for different val- For a pendulum, the Hamiltonian is ues of ω0. One immediately concludes that the larger ω0, the lesser time-dependent I(τ) becomes, which is in full agree- 1 p2 1 H = + g`ϑ 2 , (2.19) ment with (2.16). 2 `2 2 To complete the analysis, we proceed to calculate I(τ) with the asymptotic approximations for the Bessel functions as in where p = `2dϑ/dt is the generalized momentum conjugate Eq. (2.10). After a direct manipulation, we end up with to ϑ. Taking into account the relations51 H H(τ) ' √ 0 (2.25) 1 + τ d −1 −1 d −1 −1 [x J1(x)] = x J2(x), [x Y1(x)] = x Y2(x), dx dx so that (2.20) and the solution (2.7), the angular velocity of the pendulum H0 3/2 2 1/2 I(τ) ' = π`0 ϑ0 g . (2.26) can be expressed as ν0

πϑ0ω0ε √ This is an important result: for large enough values of τ, the ϑ˙ = H (2ω 1 + τ), (2.21) 1 + τ 22 0 quantity I(τ) becomes an adiabatic invariant. This validates 4 the previously suggested conclusion: the condition establish- ing the validity of the asymptotic expansion of the Bessel 0.3 functions is conceptually equivalent to the condition of adi- 0.2 abatic invariance. 0.1

III. THE EXPONENTIALLY VARYING PENDULUM 0.0

A. Basic equations of motion −0.1

−0.2 We turn now our attention to the instance where the length of the pendulum is altered non uniformly. More concretely, −0.3 we take 0 1 2 3 4 5

`(t) = ` eεt . (3.1) 0 FIG. 3. Plot of the exact solution (in red) and the asymptotic approximation (blue points) for the exponentially varying pendulum with Equation (2.2), when small oscillations are considered, re- the same parameters as in Fig. 1. duces in this case to

ϑ¨ + 2ϑ˙ + ω2e−τ ϑ = 0. (3.2) 0 can be clearly appreciated in Fig. 3, where both the exact solu- The change ϑ = θe−τ gives tion (3.7) and its asymptotic approximation (3.9) are plotted. The agreement is again remarkable. ¨ 2 −τ θ + (ω0 e − 1)θ = 0, (3.3) In contradistinction with the situation described by Eq. (2.12), the exponentially varying pendulum leads to the which has again an exact solution in terms of Bessel func- limit relation tions53. The result, employing the original variables, reads as −τ/2 lim 2ω0e = 0. (3.10) −τ −τ/2 −τ/2 τ→∞ ϑ(τ) = e [AJ2(2ω0e ) + BY2(2ω0e )]. (3.4) To fix the constants A and B we take the same initial conditions This points out the more important conceptual difference between these two cases: for the uniformly varying pendulum as before, namely ϑ(0) = ϑ0 and ϑ˙ (0) = 0. Applying the relations the validity of the asymptotic expansion of the Bessel functions only depends on the values of ε and ϑ0, but it is inde- d d [x2J (x)] = x2J (x), [x2Y (x)] = x2Y (x) (3.5) pendent on the time. For the exponentially varying pendulum dx 2 1 dx 2 1 the validity of that approximation is time dependent and for large values of the time this approximation breaks down. −τ/2 in conjunction with the change of variable x = 2ω0e and At first glance, this important difference between the two the Wronskian penduli can seems only a formal one. However, as we shall 2 see in the following it has strong implications for the adiabatic J (x)Y (x) − J (x)Y (x) = − , (3.6) 1 2 2 1 πx invariance. one can show that the final solution is −τ −τ/2 B. Adiabatic invariance ϑ(τ) = πϑ0ω0e [Y1(2ω0) J2(2ω0e ) − J ( ) Y ( e−τ/2)]. 1 2ω0 2 2ω0 (3.7) We next analyze the adiabatic change for this example. Much in the same way as for the pendulum with uniformly Equation (2.14) applied to (3.1) gives as a requirement for varying length, if adiabatic invariance

−τ/2 δ` τ/2 2π τ/2 2ω0e 1, (3.8) = εT0e = e 1, (3.11) ` ω0 is satisﬁed, we can replace equation (3.7) by its asymptotic approximation, leading to which again is formally equivalent to the condition (3.8) for the validity of the asymptotic approximation of the Bessel −3τ/4 −τ/2 functions. ϑ(τ) ' ϑ0e cos[2ω0(1 − e )]. (3.9) This indicates that even if the change of length is initially Within this approximation the maximum angular ampli- adiabatic, it will remain so only for a ﬁnite interval of time. tude ϑmax scales exactly as in Eq. (2.11), and also To put in another way, (3.11) holds true whenever limτ→∞ ϑmax(τ) = 0. However, as expected, the angular am- τ/2 τ/2 plitude ϑmax falls off more quickly in this case. This behavior ω0 e ωlim = 2πe , (3.12) 5

The function I(τ) is represented in Fig. 4. As we can see, 1.05 the ﬂuctuations of I(τ) increase with time. Thus, for large I( ) 1.00 enough time, τ will never be an invariant quantity, irrespec- tive of the value of ω0. However, for the time window chosen 0.95 in the ﬁgure, we see that for ω0 = 1000, I(τ) looks invariant over the entire interval. 0.90 Our last step is to calculate I(τ) using the asymptotic approximations for the Bessel functions. Now, we have 0.85 −τ/2 H(τ) ' H0e (3.17) 0.80 so that 0.75 0.0 0.5 1.0 1.5 2.0 H0 3/2 2 1/2 I(τ) ' = π`0 ϑ0 g , (3.18) ν0 FIG. 4. Plot of I(τ) as a function of τ for the exponentially vary- which is identical to what we have obtained for the uniformly ing pendulum with `0 = 1 m and ϑ0 = 0.3 rad. The curves corre- varying pendulum. sponds to ω0 = 2,10 and 1000 (which are associated with the values ε = 0.2491,0.0498 and 0.00005 s−1), following the decreasing am- plitudes. IV. CONCLUDING REMARKS

We have explored in detail two nontrivial yet solvable ex- ( ) or, recalling the definition of ω τ , amples of penduli of varying length with the purpose of a bet- 1 ter understanding of the concept of adiabatic invariance. τ τlim = 2ln . (3.13) The ambiguous criteria of “infinitely slow variation”, or the εT 0 assumption of “ignorance of the internal state of the system on Accordingly, there does not exist a minimum fixed value of the part of the person controlling the variable parameter”, usu- ω0 such that adiabaticity holds true forever. In our opinion ally employed to establish the condition of adiabatic change, this is the more important lesson from this paper: the adiabatic are replaced here by more quantitative criteria. condition does not need to hold, in general, for all times. For the two penduli considered in this paper, we have shown The formal equivalence discussed so far provides a math- that the physical meaning of adiabatic change is formally con- ematical interpretation fo (3.13): the change will be adia- tained in the mathematical condition of validity for the asymp- batic in those conditions in which the asymptotic expansion totic expansion of the Bessel functions: the validity of the of Bessel functions is justified. asymptotic approximation implies adiabatic change and vice Note that the change of the length in this example is in- versa. finitely continuously differentiable, but the adiabaticity holds The analysis carried out in this work invites to a more gen- true only in a time interval fixed by the values of `0 and ε. eral reflection: it is important to pay special attention to the This by no means contradict Arnold’s adiabaticity require- meaning of mathematical approximations. Actually, the ra- ment mentioned in the Introduction (that is, that the change dius of convergence of some systematic approximation to an of the length must be twice continuously differentiable), since exact solution has always a physical origin. Arnold explicitly considers a finite time interval in the definition of the adiabatic invariants5. Finally, we calculate explicitly the total energy for this case. ACKNOWLEDGMENTS Using again the Hamiltonian (2.19) and the solution (3.7) and its time derivative, we get The original ideas in this paper originated from a long co- 2 2 −τ operation with the late Richard Barakat. Over the years, they H(τ) = H0 π ω0 e have been further developed and completed with questions, 2 −τ/2 2 −τ/2 × [H11(2ω0e ) + H12(2ω0e )], (3.14) suggestions, criticism, and advice from many students and colleagues. Particular thanks for help in various ways go to with E. Bernabeu,´ J. F. Carinena,˜ A. Galindo, H. de Guise, H. Kas- H12(x) = Y1(2ω0) J2(x) − J1(2ω0) Y2(x), trup, A. B. Klimov, G. Leuchs and J. J. Monzon.´ (3.15) We are indebted to two anonymous referees for valuable comments. H11(x) = J1(2ω0) Y1(x) −Y1(2ω0) J1(x). This paper is dedicated to the memory of coauthor J. Zoido, In consequence, I(τ) becomes who unexpectedly passed away during the preparation of the final version. H0 2 2 −τ/2 2 −τ/2 2 −τ/2 This work is partially supported by the Spanish DGI I(τ) = π ω0 e [H11(2ω0e ) + H12(2ω0e )]. ν0 (Grants FIS2008-04356 and FIS2011-26786) and the UCM- (3.16) BSCH program (Grant GR-920992). 6

1 J. W. S. Rayleigh, “On the pressure of vibrations,” Phil. Mag. 3, computing,” Int. J. Mod. Phys. B 15, 1257–1286 (2001). 338–346 (1902). 27 L. Parker, “Adiabatic invariance in simple harmonic motion,” Am. 2 L. Le Cornu, “Memoire´ sur le pendule de longueur variable,” Acta J. Phys. 39, 24–27 (1971). Math. 19, 201–249 (1895). 28 M. G. Calkin, “Adiabatic invariants for varying mass,” Am. J. 3 P. Langevin and M. D. Broglie, eds., La Theorie du Rayonnement Phys. 45, 301–302 (1977). et les Quanta (Gauthier-Villars,, Paris, 1912). 29 C. Gignoux and F. Brut, “Adiabatic invariance or scaling?” Am. 4 L. Navarro and E. Perez,´ “Paul Ehrenfest: The genesis of the adia- J. Phys. 57, 422–428 (1989). batic hypothesis, 1911–1914,” Arch. Hist. Exact Sci. 60, 209–267 30 F. S. Crawford, “Elementary examples of adiabatic invariance,” (2006). Am. J. Phys. 58, 337–344 (1990). 5 V. I. Arnold, Mathematical Methods of Classical Mechanics 31 J. L. Anderson, “Multiple time scale methods for adiabatic sys- (Springer, 1978). tems,” Am. J. Phys. 60,, 923–927 (1992). 6 P. Ehrenfest, “Adiabatische Invarianten und Quantentheorie,” 32 A. C. Aguiar Pinto, M. C. Nemes, J. G. Peixoto de Faria, and Ann. Phys. (Berlin) 51, 327–352 (1916). M. T. Thomaz, “Comment on the adiabatic condition,” Am. J. 7 M. Born and V. Fock, “Beweis des Adiabatensatzes,” Z. Phys. 51, Phys. 68, 955–958 (2000). 165–180 (1928). 33 C. G. Wells and S. T. C. Siklos, “The adiabatic invariance of the 8 T. Kato, “On the adiabatic theorem of quantum mechanics,” J. action variable in classical dynamics,” Eur. J. Phys 28, 105–112 Phys. Soc. Jpn. 5, 435–439 (1950). (2007). 9 A. Sommerfeld, Atombau und Specktrallinien (Vieweg, Braun- 34 B. W. Shore, M. V. Gromovyy, L. P. Yatsenko, and V. I. Roma- schweig, 1919). nenko, “Simple mechanical analogs of rapid adiabatic passage in 10 M. Born, Vorlesungen uber¨ Atommechanik (Springer, Berlin, atomic physics,” Am. J. Phys. 77, 1183–1194 (2009). 1925). 35 K.-P. Marzlin and B. C. Sanders, “Inconsistency in the application 11 M. Jammer, The Conceptual Development of Quantum Mechanics of the adiabatic theorem,” Phys. Rev. Lett. 93, 160408 (2004). (McGraw-Hill, 1966). 36 M. S. Sarandy, L.-A. Wu, and D. Lidar, “Consistency of the adia- 12 T. Levi-Civita, “Drei Vorlesungen uber¨ adiabatische Invarianten,” batic theorem,” Quantum Inf. Process. 3, 331–349 (2004). Abh. Math. Sem. Hamburg 6, 323–366 (1928). 37 J. Du, L. Hu, Y.Wang, J. Wu, M. Zhao, and D. Suter, “Experimen- 13 T. Levi-Civita, “A general survey of the theory of adiabatic invari- tal study of the validity of quantitative conditions in the quantum ants,” J. Math. Phys. Camb. 13, 18–40 (1934). adiabatic theorem,” Phys. Rev. Lett. 101, 060403 (2008). 14 K. J. Whiteman, “Invariants and stability in classical mechanics,” 38 M. H. S. Amin, “Consistency of the adiabatic theorem,” Phys. Rep. Prog. Phys. 40, 1033–1069 (1977). Rev. Lett. 102, 220401 (2009). 15 A. Shapere and F. Wilczek, eds., Geometric Phases in Physics 39 T. Wickramasinghe and R. Ochoa, “Analysis of the linearity of (World Scientific, Singapore, 1989). half periods of the Lorentz pendulum,” Am. J. Phys. 73, 442–445 16 R. Z. Sagdeev, D. A. Usikov, and G. M. Zaslavsky, Nonlinear (2005). Physics: From The Pendulum To Turbulence And Chaos (Har- 40 A. Kavanaugh and T. Moe, “The pit and the pendulum,” wood, New York, 1988). http://online.redwoods.cc.ca.us/instruct/darnold/deproj/sp05/atrav/ThePitandThePendulum.pdf 17 P. Lochak and C. Meunier, Multiphase Averaging for Classi- (2005). cal Systems. With Applications to Adiabatic Theorems (Springer, 41 G. Krutkow and V. Fock, “Uber¨ das Rayleighesche Pendel,” Z. New York, 1988). Phys. 13, 195–202 (1923). 18 J. Oreg, F. T. Hioe, and J. H. Eberly, “Adiabatic following in mul- 42 R. M. Kulsrud, “Adiabatic invariant of the harmonic oscillator,” tilevel systems,” Phys. Rev. A 29, 690–697 (1984). Phys. Rev. 106, 205–207 (1957). 19 U. Gaubatz, P. Rudecki, M. Becker, S. Schiemann, M. Kulz,¨ and 43 C. S. Gardner, “Adiabatic invariants of periodic classical sys- K. Bergmann, “Population switching between vibrational levels tems,” Phys. Rev. 115, 791–794 (1959). in molecular beams,” Chem. Phys. Lett. 149, 463–468 (1988). 44 M. Kruskal, “Asymptotic theory of Hamiltonian and other sys- 20 U. Gaubatz, P. Rudecki, S. Schiemann, and K. Bergmann, “Popu- tems with all solutions nearly periodic,” J. Math. Phys. 3, 806– lation transfer between molecular vibrational levels by stimulated 828 (1962). Raman scattering with partially overlapping laser fields. A new 45 J. E. Littlewood, “Lorentz’s pendulum problem,” Ann. Phys. 21, concept and experimental results,” J. Chem. Phys. 92, 5363–5376 233–242 (1963). (1990). 46 M. N. Brearley, “The simple pendulum with uniformly changing 21 S. Schiemann, A. Kuhn, S. Steuerwald, and K. Bergmann, “Effi- string length,” Proc. Edin. Math. Soc. 15, 61–66 (1966). cient coherent population transfer in NO molecules using pulsed 47 A. Werner and C. J. Eliezer, “The lengthening pendulum,” J. Aust. lasers,” Phys. Rev. Lett. 71, 3637–3640 (1993). Math. Soc. 9, 331–336 (1969). 22 P. Pillet, C. Valentin, R. L. Yuan, and J. Yu, “Adiabatic population 48 D. K. Ross, “The behaviour of a simple pendulum with uni- transfer in a multilevel system,” Phys. Rev. A 48, 845–848 (1993). formly shortening string length,” Int. J. Nonlin. Mech. 14, 175– 23 P. Kral,´ I. Thanopulos, and M. Shapiro, “Colloquium: Coherently 182 (1979). controlled adiabatic passage,” Rev. Mod. Phys. 79, 53–77 (2007). 49 H. Goldstein, Classical Mechanics (Addison-Wesley, New York, 24 E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser, “Quantum 1980). computation by adiabatic evolution,” . 50 J. V. Jose´ and E. J. Saletan, Classical Dynamics: A Contemporary 25 E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, and Approach (Cambridge University Press, Cambridge, 1998). D. Preda, “A quantum adiabatic evolution algorithm applied to 51 N. W. McLachlan, Bessel Functions for Engineers (Oxford Uni- random instances of an np-complete problem,” Science 292, 472– versity Press, Oxford, 1955). 475 (2001). 52 C. Andrade, The Structure of the Atom (Harcourt, New York, 26 J. Pachos and P. Zanardi, “Quantum holonomies for quantum 1962). 7

53 E. Kamke, Differentialgleichungen: Losungsmethoden¨ und Losungen¨ , vol. 1 (Chelsea, New York, 1974).