We analyze a simple variant of the Lorentz pendulum in which the length is varied exponentially instead of uniformly as in the standard case. We establish quantitative criteria for the condition of adiabatic changes in both pendula and demonstrate their substantially different physical behavior with regard to adiabatic invariance.
REFERENCES
1.
J. W. S.
Rayleigh
, “On the pressure of vibrations
,” Philos. Mag.
3
, 338
–346
(1902
).2.
L.
Le Cornu
, “Mémoire sur le pendule de longueur variable
,” Acta Math.
19
, 201
–249
(1895
).3.
La Theorie du Rayonnement et les Quanta
, edited by P.
Langevin
and M. D.
Broglie
(Gauthier-Villars
, Paris
, 1912
).4.
L.
Navarro
and E.
Pérez
, “Paul Ehrenfest: The genesis of the adiabatic hypothesis, 1911–1914
,” Arch. Hist. Exact Sci.
60
, 209
–267
(2006
).5.
V. I.
Arnold
, Mathematical Methods of Classical Mechanics
(Springer
, New York
, 1978
).6.
P.
Ehrenfest
, “Adiabatische Invarianten und Quantentheorie
,” Ann. Phys. (Berlin)
356
, 327
–352
(1916
).7.
M.
Born
and V.
Fock
, “Beweis des Adiabatensatzes
,” Z. Phys.
51
, 165
–180
(1928
).8.
T.
Kato
, “On the adiabatic theorem of quantum mechanics
,” J. Phys. Soc. Jpn.
5
, 435
–439
(1950
).9.
10.
11.
M.
Jammer
, The Conceptual Development of Quantum Mechanics
(McGraw-Hill
, New York
, 1966
).12.
T.
Levi-Civita
, “Drei Vorlesungen über adiabatische Invarianten
,” Abh. Math. Sem. Hamburg
6
, 323
–366
(1928
).13.
T.
Levi-Civita
, “A general survey of the theory of adiabatic invariants
,” J. Math. Phys.
13
, 18
–40
(1934
).14.
K. J.
Whiteman
, “Invariants and stability in classical mechanics
,” Rep. Prog. Phys.
40
, 1033
–1069
(1977
).15.
Geometric Phases in Physics
, edited by A.
Shapere
and F.
Wilczek
(World Scientific
, Singapore
, 1989
).16.
R. Z.
Sagdeev
, D. A.
Usikov
, and G. M.
Zaslavsky
, Nonlinear Physics: From the Pendulum to Turbulence and Chaos
(Harwood
, New York
, 1988
).17.
P.
Lochak
and C.
Meunier
, Multiphase Averaging for Classical Systems. With Applications to Adiabatic Theorems
(Springer
, New York
, 1988
).18.
J.
Oreg
, F. T.
Hioe
, and J. H.
Eberly
, “Adiabatic following in multilevel systems
,” Phys. Rev. A
29
, 690
–697
(1984
).19.
U.
Gaubatz
, P.
Rudecki
, M.
Becker
, S.
Schiemann
, M.
Külz
, and K.
Bergmann
, “Population switching between vibrational levels in molecular beams
,” Chem. Phys. Lett.
149
, 463
–468
(1988
).20.
U.
Gaubatz
, P.
Rudecki
, S.
Schiemann
, and K.
Bergmann
, “Population transfer between molecular vibrational levels by stimulated Raman scattering with partially overlapping laser fields. A new concept and experimental results
,” J. Chem. Phys.
92
, 5363
–5376
(1990
).21.
S.
Schiemann
, A.
Kuhn
, S.
Steuerwald
, and K.
Bergmann
, “Efficient coherent population transfer in NO molecules using pulsed lasers
,” Phys. Rev. Lett.
71
, 3637
–3640
(1993
).22.
P.
Pillet
, C.
Valentin
, R. L.
Yuan
, and J.
Yu
, “Adiabatic population transfer in a multilevel system
,” Phys. Rev. A
48
, 845
–848
(1993
).23.
P.
Král
, I.
Thanopulos
, and M.
Shapiro
, “Colloquium: Coherently controlled adiabatic passage
,” Rev. Mod. Phys.
79
, 53
–77
(2007
).24.
E.
Farhi
, J.
Goldstone
, S.
Gutmann
, and M.
Sipser
, “Quantum computation by adiabatic evolution
,” e-print arXiv quant-ph/0001106.25.
E.
Farhi
, J.
Goldstone
, S.
Gutmann
, J.
Lapan
, A.
Lundgren
, and D.
Preda
, “A quantum adiabatic evolution algorithm applied to random instances of an np-complete problem
,” Science
292
, 472
–475
(2001
).26.
J.
Pachos
and P.
Zanardi
, “Quantum holonomies for quantum computing
,” Int. J. Mod. Phys. B
15
, 1257
–1286
(2001
).27.
L.
Parker
, “Adiabatic invariance in simple harmonic motion
,” Am. J. Phys.
39
, 24
–27
(1971
).28.
M. G.
Calkin
, “Adiabatic invariants for varying mass
,” Am. J. Phys.
45
, 301
–302
(1977
).29.
C.
Gignoux
and F.
Brut
, “Adiabatic invariance or scaling?
,” Am. J. Phys.
57
, 422
–428
(1989
).30.
F. S.
Crawford
, “Elementary examples of adiabatic invariance
,” Am. J. Phys.
58
, 337
–344
(1990
).31.
J. L.
Anderson
, “Multiple time scale methods for adiabatic systems
,” Am. J. Phys.
60
, 923
–927
(1992
).32.
A. C.
Aguiar Pinto
, M. C.
Nemes
, J. G.
Peixoto de Faria
, and M. T.
Thomaz
, “Comment on the adiabatic condition
,” Am. J. Phys.
68
, 955
–958
(2000
).33.
C. G.
Wells
and S. T. C.
Siklos
, “The adiabatic invariance of the action variable in classical dynamics
,” Eur. J. Phys
28
, 105
–112
(2007
).34.
B. W.
Shore
, M. V.
Gromovyy
, L. P.
Yatsenko
, and V. I.
Romanenko
, “Simple mechanical analogs of rapid adiabatic passage in atomic physics
,” Am. J. Phys.
77
, 1183
–1194
(2009
).35.
A. C.
Lobo
, R. A.
Ribeiro
, C.
de Assis Ribeiro
, and P. R.
Dieguez
, “Geometry of the adiabatic theorem
,” Eur. J. Phys
33
, 1063
–1072
(2012
).36.
K.-P.
Marzlin
and B. C.
Sanders
, “Inconsistency in the application of the adiabatic theorem
,” Phys. Rev. Lett.
93
, 160408
(2004
).37.
M. S.
Sarandy
, L.-A.
Wu
, and D.
Lidar
, “Consistency of the adiabatic theorem
,” Quantum Inf. Process.
3
, 331
–349
(2004
).38.
J.
Du
, L.
Hu
, Y.
Wang
, J.
Wu
, M.
Zhao
, and D.
Suter
, “Experimental study of the validity of quantitative conditions in the quantum adiabatic theorem
,” Phys. Rev. Lett.
101
, 060403
(2008
).39.
M. H. S.
Amin
, “Consistency of the adiabatic theorem
,” Phys. Rev. Lett.
102
, 220401
(2009
).40.
T.
Wickramasinghe
and R.
Ochoa
, “Analysis of the linearity of half periods of the Lorentz pendulum
,” Am. J. Phys.
73
, 442
–445
(2005
).41.
A.
Kavanaugh
and T.
Moe
, “The pit and the pendulum,” <http://online.redwoods.cc.ca.us/instruct/darnold/deproj/sp05/atrav/ThePitandThePendulum.pdf> (2005
).42.
G.
Krutkow
and V.
Fock
, “Über das Rayleighesche Pendel
,” Z. Phys.
13
, 195
–202
(1923
).43.
R. M.
Kulsrud
, “Adiabatic invariant of the harmonic oscillator
,” Phys. Rev.
106
, 205
–207
(1957
).44.
C. S.
Gardner
, “Adiabatic invariants of periodic classical systems
,” Phys. Rev.
115
, 791
–794
(1959
).45.
M.
Kruskal
, “Asymptotic theory of Hamiltonian and other systems with all solutions nearly periodic
,” J. Math. Phys.
3
, 806
–828
(1962
).46.
J. E.
Littlewood
, “Lorentz's pendulum problem
,” Ann. Phys.
21
, 233
–242
(1963
).47.
M. N.
Brearley
, “The simple pendulum with uniformly changing string length
,” Proc. Edin. Math. Soc.
15
, 61
–66
(1966
).48.
A.
Werner
and C. J.
Eliezer
, “The lengthening pendulum
,” J. Aust. Math. Soc.
9
, 331
–336
(1969
).49.
D. K.
Ross
, “The behaviour of a simple pendulum with uniformly shortening string length
,” Int. J. Nonlin. Mech.
14
, 175
–182
(1979
).50.
51.
J. V.
José
and E. J.
Saletan
, Classical Dynamics: A Contemporary Approach
(Cambridge U.P.
, Cambridge
, 1998
).52.
53.
54.
E.
Kamke
, Differentialgleichungen: Lösungsmethoden und Lösungen
, Vol. 1 (Chelsea
, New York
, 1974
).© 2013 American Association of Physics Teachers.
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