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From the history of physics


Topological phase in classical mechanics

 a,  b
a Federal Research Center A.V. Gaponov-Grekhov Institute of Applied Physics of the Russian Academy of Sciences, ul. Ulyanova 46, Nizhny Novgorod, 603000, Russian Federation
b V. I. Kuznetsov Federal State Unitary Enterprise Research Institute of Applied Mechanics, ul. Aviamotornaya 55, Moscow, 111123, Russian Federation

The historical development of the concept of the topological phase in classical mechanics from the mid-19th century to the present is discussed. There are three stages to be recognized in this period. The first, the mid-19th century stage, is concerned with studying the kinematics of rigid body rotation and includes such milestone developments as the Euler theorem on finite rotation of rigids, Gauss formula for the sum excess of the angles of a spherical polygon, Rodrigues’s proof of the noncommutativity property of two finite rotations, and, finally, Hamilton’s Lectures on Quaternions where the solid angle theorem is formulated and proved. The experimental discovery of the nonholonomic error of gyroscopes and its exhaustive explanation by A Yu Ishlinskii represent the second stage. The third stage, which started in the 1980s, has witnessed the rediscovery of the nonholonomic effect in the framework of Hamiltonian formalism and is dominated by the study of how the topological phase — or an additional angle — forms in a mechanical system being treated in action-angle variables.

Fulltext pdf (317 KB)
Fulltext is also available at DOI: 10.1070/PU2003v046n09ABEH001635
PACS: 01.65.+g, 03.65.Vf, 45.40.Cc (all)
DOI: 10.1070/PU2003v046n09ABEH001635
URL: https://ufn.ru/en/articles/2003/9/d/
000188171400004
Citation: Malykin G B, Kharlamov S A "Topological phase in classical mechanics" Phys. Usp. 46 957–965 (2003)
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Оригинал: Малыкин Г Б, Харламов С А «Топологическая фаза в классической механике» УФН 173 985–994 (2003); DOI: 10.3367/UFNr.0173.200309e.0985

References (85) ↓ Cited by (15) Similar articles (20)

  1. Berry M V Proc. R. Soc. London Ser. A 392 45 (1984)
  2. Simon B Phys. Rev. Lett. 51 2167 (1983)
  3. Vinitskii S I et al. Usp. Fiz. Nauk 160 (6) 1 (1990) [Sov. Phys. Usp. 33 403 (1990)]
  4. Berry M Phys. Today 43 (12) 34 (1990)
  5. Anandan J Nature 360 307 (1992)
  6. Klyshko D N Usp. Fiz. Nauk 163 (11) 1 (1993) [Phys. Usp. 36 1005 (1993)]
  7. Toronov V Yu, Derbov V L, Priyutova O M Izv. Vyssh. Uchebn. Zaved. Prikladnaya Nelineinaya Dinamika 4 (6) 3 (1996)
  8. Bodnarchuk V I, Davtyan L S, Korneev D A Usp. Fiz. Nauk 166 185 (1996) [Phys. Usp. 39 169 (1996)]
  9. Malykin G B Izv. Vyssh. Uchebn. Zaved. Radiofiz. 40 265 (1997) [Radiophys. Quantum Electron. 40 175 (1997)]
  10. Berry M Nature 326 277 (1987)
  11. Rytov S M Dokl. Akad. Nauk SSSR 18 263 (1938)
  12. Rytov S M Tr. Fiz. Inst. Akad. Nauk SSSR 2 (1) 41 (1940)
  13. Vladimirskii V V Dokl. Akad. Nauk SSSR 31 222 (1941)
  14. Pancharatnam S Proc. Ind. Acad. Sci. A 44 247 (1956)
  15. Pancharatnam S Proc. Ind. Acad. Sci. A 44 398 (1956)
  16. Berry M V J. Mod. Optics 34 1400 (1987)
  17. Hannay J H J. Phys. A: Math. Gen. 18 221 (1985)
  18. Hamilton W R Lectures on Quaternions (Dublin: Hodges and Smith, 1853) p. 338
  19. Lamb H Higher Mechanics (Cambridge: The Univ. Press, 1929) [Translated into Russian (Moscow-Leningrad: ONTI, 1936)]
  20. Euler L Theoria motus corporum solidorum seu rigidorum (Rostock: Greifswald, 1765)
  21. Burago Yu D, in Matematicheskaya Entsiklopediya (Mathematical Encyclopedia) Vol. 1 (Moscow: Sov. Entsiklopediya, 1977) p. 902
  22. Rodrigues O J. Math. Pures Appl. 5 380 (1840)
  23. Donkin M J Philos. Mag. 4 1 (1851)
  24. Hamilton W R Trans. R. Irish Acad. 15 69 (1828); 16 (1) 1 (1830); 16 (2) 93 (1831); 17 (1) 1 (1837) [Translated into Russian: Izbrannye Trudy (Moscow: Nauka, 1994) pp.10 – 166]
  25. C.H.D.Z. Traité de la lumiére (Leyden, 1690) [Translated into Russian (Moscow-Leningrad: ONTI, 1935)]
  26. Klein F, in Gesammelte Mathematische Abhandlungen Bd. 2 (Hrsg. R Fricke, H Vermeil, Berlin: J. Springer, 1922) p. 601
  27. Chetaev N G Zh. Priklad. Matem. Mekh. 22 487 (1958) [J. Appl. Math. Mekh. 22 678 (1959)]; also in Ustoichivost’ Dvizheniya. Raboty po Analiticheskoi Dinamike (Stability of Motion. Works on Analytical Mechanics, Moscow: Izd. AN SSSR, 1962) p. 393
  28. Chetaev N G Zh. Priklad. Matem. Mekh. 24 23 (1960) [J. Appl. Math. Mech. 24 27 (1960)]; also in Ustoichivost’ Dvizheniya. Raboty po Analiticheskoi Dinamike (Stability of Motion. Works on Analytical Mechanics, Moscow: Izd. AN SSSR, 1962) p. 404
  29. Emel’yanova I S, in Matematicheskoe Modelirovanie i Optimal’noe Upravlenie (Mathematical Modeling and Optimal Control, Ed. S N Strongin, Nizhnii Novgorod: Izd. NNGU, 1994) p. 90
  30. Emel’yanova I S, Dunaeva N L, in Problemy Teorii Kolebanii (Problems of the Theory of Vibrations)(Ed. E F Sabaev, Nizhnii Novgorod: Izd. NNGU, 1995) p. 3
  31. Emel’yanova I S, in Metody Analiza Nelineinykh Sistem (Methods of Analysis for Nonlinear Systems, Eds S V Emel’yanov, S K Korovin, Moscow: Dialog MGU, 1997) p. 63
  32. Pauli W Helv. Phys. Acta 5 179 (1932) [Translated into Russian: Trudy po Kvantovoi Mekhanike, Stat’i 1928 – 1938 (Moscow: Nauka, 1977) p. 112]
  33. Galanin A D J. Phys. USSR 6 35 (1942)
  34. Thomas L H Nature 117 514 (1926)
  35. Thomas L H Philos. Mag. Ser. 7 3 (13) 1 (1927)
  36. Ishlinskii A Yu Priborostroenie (4) 3 (1944)
  37. Ishlinskii A Yu Mekhanika Spetsial’nykh Giroskopicheskikh Sistem (Mechanics of Special Gyroscopic Systems, Kiev: Izd. AN UkrSSR, 1952)
  38. Ishlinskii A Yu Mekhanika Giroskopicheskikh Sistem (Mechanics of Gyroscopic Systems) 2nd ed. (Moscow: Izd. AN SSSR, 1963) [Translated into English (Jerusalem: Israel Program for Scientific Translations, 1965)]
  39. Ishlinskii A Yu, Borzov V I, Stepanenko N P Lektsii po Teorii Giroskopov (Lectures on the Theory of Gyroscopes, Moscow: Izd. MGU, 1983)
  40. Foucault L C.R. Acad. Sci. 35 421 (1852)
  41. Bulgakov B V Prikladnaya Teoriya Giroskopov (Applied Theory of Gyroscopes, Moscow-Leningrad: GITTL, 1939)
  42. Krylov A N, Krutkov Yu F Obshchaya Teoriya Giroskopov (General Theory of Gyroscopes, Leningrad: Izd. AN SSSR, 1932)
  43. Ishlinskii A Yu Orientatsiya, Giroskopy i Inertsial’naya Navigatsiya (Orientation, Gyroscopes, and Inertial Navigation, Moscow: Nauka, 1976)
  44. Ishlinskii A Yu Priborostroenie (1) 12 (1944)
  45. Rashevskii P K Rimanova Geometriya i Tenzornyi Analiz (Riemannian Geometry and Tensor Calculus) 3rd ed. (Moscow: Nauka, 1967)
  46. Lumiste Yu G, in Matematicheskaya Entsiklopediya (Mathematical Encyclopedia) Vol. 4 (Moscow: Sov. Entsiklopediya, 1984) p. 206
  47. Neimark Yu I, Fufaev N A Dinamika Negolonomnykh Sistem (Dynamics of Nonholonomic Systems, Moscow: Nauka, 1967) [Translated into English (Providence, RI: American Mathematical Soc., 1972)]
  48. Heinrich G Österr. Ing. Arch. 3 215 (1950)
  49. Magnus K Kreisel. Theorie und Anwendungen (Berlin: Springer-Verlag, 1971) [Translated into Russian (Moscow: Mir, 1974)]
  50. Price H L Aircraft Eng. 20 (1) 11 (1948)
  51. Price H L Aircraft Eng. 20 (2) 38 (1948)
  52. Goodman L E, Robinson A R J. Appl. Mech. 25 210 (1958) [Translated into Russian, in Mekhanika (Mechanics) No. 5 (57, Moscow: IL, 1959) p. 133]
  53. Berry M V, Hannay J H J. Phys. A: Math. Gen. 21 L325 (1988)
  54. Arnol’d V I Matematicheskie Metody Klassicheskoi Mekhaniki (Mathematical Methods of Classical Mechanics, Moscow: Nauka, 1974) [Translated into English (New York: Springer-Verlag, 1978)]
  55. Andronov A A, Mandel’shtam L I, Leontovich M A Zh. Russ. Fiz. Khim. Obshch. Ch. Fiz. 60 413 (1928); see also Andronov A A Sobranie Trudov (Collected Works, Moscow: Izd. AN SSSR, 1956) p. 34; Mandel’shtam L I Polnoe Sobranie Trudov (Complete Works) Vol. 1 (Leningrad: Izd. AN SSSR, 1948) p. 297
  56. Zhbanov Yu K, Zhuravlev V F Izv. Akad. Nauk SSSR. Ser. Mekh. Tverd. Tela (1) 9 (1978)
  57. Zhuravlev V F Zh. Prikl. Mat. Mekh. 60 323 (1996) [J. Appl. Math. Mekh. 60 319 (1996)]
  58. Zhuravlev V F Osnovy Teoreticheskoi Mekhaniki (Fundamentals of Theoretical Mechanics, Moscow: Nauka – Fizmatlit, 1997)
  59. Krobka N I, in Proc. of the 2nd Saint Petersburg Intern. Conf. on Gyroscopic Technology and Navigation Pt. II (St. Petersburg, 1995) p. 99
  60. Perelyaev S E, in Mekhanika i Navigatsiya. Materialy Nauchnoi Sessii, Posvyashchennoi 85-Letiyu Akademika RAN A Yu Ishlinskogo, Moskva, 30 Sent. 1998 (Mechanics and Navigation. Reports of the Scientific Session Celebrating the 85th Anniversary of Academician A Yu Ishlinskii’s Birth, Moscow, Sept. 30, 1998, Ed Yu M Kolesov, St. Petersburg: GNTs RF — TsNII Elektropribor, 1999) p. 88
  61. Krutov A V Izv. Vyssh. Uchebn. Zaved. Mashinostroenie (4) 13 (2001)
  62. Simon R, Mukunda N, Sudarshan E C G Phys. Rev. Lett. 62 1331 (1989)
  63. Aharonov Y, Anandan J Phys. Rev. Lett. 58 1593 (1987)
  64. Song D-Y Phys. Rev. Lett. 85 1141 (2000)
  65. Malykin G B, Neimark Yu I Izv. Vyssh. Uchebn. Zaved. Radiofiz. 41 1125 (1998)
  66. Malykin G B Opt. i Spektrosk. 81 474 (1996) [Opt. Spectrosc. 81 431 (1996)]
  67. Malykin G B Opt. i Spektrosk. 84 515 (1998) [Opt. Spectrosc. 84 455 (1998)]
  68. Andronova I A, Gelikonov G V, Malykin G B Kvantovaya Elektron. 26 271 (1999) [Quantum Electron. 26 271 (1999)]
  69. Andronova I A, Gelikonov G V, Malykin G B Proc. SPIE 3736 423 (1999)
  70. Andronova I A, Malykin G B Usp. Fiz. Nauk 172 849 (2002) [Phys. Usp. 45 793 (2002)]
  71. Shurkliff W A Polarized Light (Cambridge, Mass.: Harvard Univ. Press, 1962) [Translated into Russian (Moscow: Mir, 1965)]
  72. Logunov A A, Chugreev Yu V Usp. Fiz. Nauk 156 137 (1988) [Sov. Phys. Usp. 31 861 (1988)]
  73. Malykin G B Usp. Fiz. Nauk 167 337 (1997) [Phys. Usp. 40 317 (1997)]
  74. Vugal’ter G A, Malykin G B Izv. Vyssh. Uchebn. Zaved. Radiofiz. 42 373 (1999)
  75. Malykin G B Usp. Fiz. Nauk 170 1325 (2000) [Phys. Usp. 43 1229 (2000)]
  76. Malykin G B Usp. Fiz. Nauk 172 969 (2002) [Phys. Usp. 45 907 (2002)]
  77. Malykin G B Usp. Fiz. Nauk 169 585 (1999) [Phys. Usp. 42 505 (1999)]
  78. Malykin G B Zh. Prikl. Mat. Mekh. 63 775 (1999) [J. Appl. Math. Mech. 63 731 (1999)]
  79. Malykin G B Izv. Ross. Akad. Nauk Ser. Mekh. Tverd. Tela (4) 187 (2000)
  80. Malykin G B, Permitin G V, in Fizicheskaya Entsiklopediya (Physical Encyclopedia) Vol. 5 (Ed.-in-Chief A M Prokhorov, Moscow: Bol’shaya Rossiiskaya Entsiklopediya, 1998) p. 123
  81. Penrose R Proc. Cambr. Philos. Soc. 55 (1) 137 (1959)
  82. Terrell J Phys. Rev. 116 1041 (1959)
  83. Weisskopf V F Physics in the Twentieth Century (Cambridge, Mass.: MIT Press, 1972) [Translated into Russian (Moscow: Atomizdat, 1977) p. 179]
  84. Bolotovskii B M, in Einshteinovskii Sbornik 1986 – 1990 (Einstein’s Collection 1986 – 1990, Ed. I Yu Kobzarev, Moscow: Nauka, 1990) p. 279
  85. Burlankov D E, Malykin G B, Preprint No. 576 (Nizhnii Novgorod: Institute of Applied Physics of the Russian Academy of Sciences, 2001)
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