RSS feeds
РусскийEnglish
Issue 11, 2024
Volume year page
Issues Authors PACS Subscription For authors

Issues

 / 

2010

 / 

September

←  →

From the history of physics


Noncommutative nature of the addition of noncollinear velocities in special relativity and the geometric phase method (commemorating the publication centennial of A Sommerfeld’s work)


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

In 1909, Arnold Sommerfeld used geometric calculations to show that the relativistic addition of two noncollinear velocities on an imaginary-radius sphere is a noncommutative operation. Sommerfeld was the first to use the geometric phase method to calculate the angle between the resulting velocities depending on the order in which they are added. For this, he related the value of this angle to the excess of the spherical triangle formed by the two original velocities and their sum. In 1931, Sommerfeld applied his method to analyze the Thomas precession.

This article was written to commemorate the publication centennial of A Sommerfeld’s paper “Über die Zusammensetzung der Geschwindigkeiten in der Relativtheorie” (On the addition of velocities in relativity theory) in Physikalische Zeitschrift.

Fulltext pdf (122 KB)
Fulltext is also available at DOI: 10.3367/UFNe.0180.201009d.0965
PACS: 01.65.+g, 03.30.+p, 03.65.Vf (all)
DOI: 10.3367/UFNe.0180.201009d.0965
URL: https://ufn.ru/en/articles/2010/9/d/
000286041500004
2010PhyU...53..923M
Citation: Malykin G B "Noncommutative nature of the addition of noncollinear velocities in special relativity and the geometric phase method (commemorating the publication centennial of A Sommerfeld's work)" Phys. Usp. 53 923–927 (2010)
BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Оригинал: Малыкин Г Б «Некоммутативность сложения неколлинеарных скоростей в специальной теории относительности и метод геометрической фазы (к столетию со дня публикации работы А. Зоммерфельда)» УФН 180 965–969 (2010); DOI: 10.3367/UFNr.0180.201009d.0965

A. Sommerfeld is author of Physics-Uspekhi

References (65) ↓ Cited by (2) Similar articles (20)

  1. Sommerfeld A Phys. Z. 10 826 (1909)
  2. Sommerfeld A Verhandlungen Deutschen Phys. Gesellschaft 11 577 (1909)
  3. Sommerfeld A Atombau Und Spektrallinien Vol. 1 (Braunshweig: F. Vieweg & Sohn, 1931) p. 707; Zommerfel’d A Stroenie Atoma i Spektry Vol. 1 (M.: GITTL, 1956)
  4. Thomas L H Nature 117 514 (1926)
  5. Thomas L H Philos. Mag. Ser. 7 3 1 (1927)
  6. Malykin G B, Permitin G V "Tomasovskaya pretsessiya" Fizicheskaya Entsiklopediya Vol. 5 (M.: Bol’shaya Ros. entsiklopediya, 1998)
  7. Malykin G B Usp. Fiz. Nauk 169 585 (1999); Malykin G B Phys. Usp. 42 505 (1999)
  8. Malykin G B Priklad. Mat. Mekh. 63 775 (1999); Malykin G B J. Appl. Math. Mech. 63 731 (1999)
  9. Malykin G B Usp. Fiz. Nauk 176 865 (2006); Malykin G B Phys. Usp. 49 837 (2006)
  10. Ritus V I Usp. Fiz. Nauk 177 105 (2007); Ritus V I Phys. Usp. 50 95 (2007)
  11. Ritus V I Usp. Fiz. Nauk 178 739 (2008); Ritus V I Phys. Usp. 51 709 (2008)
  12. Belloni L, Reina C Eur. J. Phys. 7 55 (1986); Belloni L, Reina Ch Einshteinovskii Sbornik. 1984 - 1985 (Pod red. I Yu Kobzareva) (M.: Nauka, 1988) p. 201
  13. Walter S "The non-Euclidean style of Minkowskian relativity" The Symbolic Universe: Geometry And Physics 1890 - 1930 (Ed. J Gray) (Oxford: Oxford Univ. Press, 1999) p. 91; Walter S http://www.univ-nancy2.fr/DepPhilo/walter/papers/nes.pdf
  14. Vinitskii S I i dr. Usp. Fiz. Nauk 160 (6) 1 (1990); Vinitskii S I et al. Sov. Phys. Usp. 33 403 (1990)
  15. Berry M Phys. Today 43 (12) 34 (1990)
  16. Anandan J Nature 360 307 (1992)
  17. Klyshko D N Usp. Fiz. Nauk 163 (11) 1 (1993); Klyshko D N Phys. Usp. 36 1005 (1993)
  18. Bodnarchuk V I, Davtyan L S, Korneev D A Usp. Fiz. Nauk 166 185 (1996); Bodnarchuk V I, Davtyan L S, Korneev D A Phys. Usp. 39 169 (1996)
  19. Malykin G B Optika Spektroskopiya 81 474 (1996); Malykin G B Opt. Spectrosc. 81 431 (1996)
  20. Malykin G B Izv. Vuzov. Radiofizika 40 265 (1997); Malykin G B Radiophys. Quantum Electron. 40 175 (1997)
  21. Malykin G B Optika Spektroskopiya 84 515 (1998); Malykin G B Opt. Spectrosc. 84 455 (1998)
  22. Malykin G B, Kharlamov S A Usp. Fiz. Nauk 173 985 (2003); Malykin G B, Kharlamov S A Phys. Usp. 46 957 (2003)
  23. Malykin G B, Pozdnyakova V I Usp. Fiz. Nauk 174 303 (2004); Malykin G B, Pozdnyakova V I Phys. Usp. 47 289 (2004)
  24. Hamilton W R Lectures On Quaternions (Dublin: Hodges and Smith, 1853) p. 338-340
  25. Larmor J Aether And Matter (Cambridge: Univ. Press, 1900); Larmor Dzh "Efir i materiya" Printsip Otnositel’nosti (Sost. A A Tyapkin) (M.: Atomizdat, 1973) p. 48
  26. Einstein A Ann. Physik 17 891 (1905); Einshtein A Sobranie Sochinenii Vol. 1 (M.: Nauka, 1965) p. 7
  27. Poincaré H Palermo Rend. 21 129 (1906); Puankare A Izbrannye Trudy Vol. 3 (M.: Nauka, 1974) p. 433
  28. Fizeau H C.R. Acad. Sci. Paris 33 349 (1851); Fizo I Tvortsy Fizicheskoi Optiki (Sost. U I Frankfurt) (M.: Nauka, 1973) p. 214
  29. Malykin G B Usp. Fiz. Nauk 170 1325 (2000); Malykin G B Phys. Usp. 43 1229 (2000)
  30. Minkowski G Phys. Z. 10 104 (1909); Minkovskii G Printsip Otnositel’nosti (Pod red V K Frederiksa, D D Ivanenko) (L.: ONTI, 1935) p. 181; Minkovskii G Printsip Otnositel’nosti (Sost. A A Tyapkin) (M.: Atomizdat, 1973) p. 167
  31. Crantz P Sphärische Trigonometrie (Leipzig: B.G. Teubner, 1920); Krants P Sfericheskaya Trigonometriya (Berlin: I.P. Ladyzhnikov, 1923); Krants P Sfericheskaya Trigonometriya 2-e izd. (M.: URSS, 2007)
  32. Sommerfeld A Ann. Physik 337 749 (1910)
  33. Sommerfeld A Ann. Physik 338 649 (1910)
  34. Varičak V Phys. Z. 11 93 (1910)
  35. Varičak V Phys. Z. 11 287 (1910)
  36. Varičak V Phys. Z. 11 586 (1910)
  37. Varičak V Jayhresbericht Der Deutschen Mathematiker-Vereinigung 21 103 (1912)
  38. Laue M Das Relativitätprinzip (Braunschweig: F. Vieweg & Sohn, 1911)
  39. Wilson E B, Lewis G N Proc. Am. Acad. Arts Sci. 48 387 (1912)
  40. Robb A A Optical Geometry Of Motion: A New View Of The Theory Of Relativity (Cambridge: W. Heffer and Sons, 1911)
  41. Borel É C.R. Acad. Sci. Paris 156 215 (1913)
  42. Borel É C.R. Acad. Sci. Paris 157 703 (1913)
  43. Borel É Introduction Géométrique (Paris: Gauthier-Villar, 1914)
  44. Ogura K Tóhoku Univ. Sci. Rep. 2 95 (1913)
  45. Föppl L, Daniell P Gött. Nachr. (4) 519 (1913)
  46. Silberstein L The Theory Of Relativity (London: Macmillan and Co., 1914)
  47. Smorodinskii Ya A Zh. Eksp. Teor. Fiz. 43 2217 (1962); Smorodinskij Ya A Sov. Phys. JETP 16 1566 (1963)
  48. Smorodinskii Ya A Atomnaya Energiya 14 110 (1963); Smorodinskii Ya A Atom. Energ. 14 102 (1963)
  49. Permitin G V, Smirnov A I Zh. Eksp. Teor. Fiz. 119 16 (2001); Permitin G V, Smirnov A I JETP 92 10 (2001)
  50. Rhodes J A, Semon M D Am. J. Phys. 72 943 (2004)
  51. Ishlinskii A Yu Mekhanika Spetsial’nykh Giroskopicheskikh Sistem (Kiev: Izd-vo AN USSR, 1952)
  52. Goudsmit S, Uhlenbeck G E Physica 5 266 (1925)
  53. Sommerfeld A Atti Dei Convegni Di Fisica Nucleare (Roma, 1931, 1932)
  54. Møller C The Theory Of Relativity (1972); Meller K Teoriya Otnositel’nosti (M.: Atomizdat, 1975)
  55. Ritus V I Zh. Eksp. Teor. Fiz. 40 352 (1961); Ritus V I Sov. Phys. JETP 13 240 (1961)
  56. Jackson J D Classical Electrodynamics (New York: Wiley, 1962); Dzhekson Dzh Klassicheskaya Elektrodinamika (M.: Mir, 1965)
  57. Chakrabarti A J. Math. Phys. 5 1747 (1964)
  58. Rytov S M Dokl. Akad. Nauk SSSR 18 263 (1938); Rytov S M C.R. (Dokl.) Acad. Sci. USSR 18 263 (1938)
  59. Vladimirskii V V Dokl. Akad. Nauk SSSR 31 222 (1941)
  60. Galanin A D J. Phys. USSR 6 35 (1942)
  61. Ishlinskii A Yu Priborostroenie. Tekhn.-proizv. Byull. (4) 3 (1944)
  62. Pancharatnam S Proc. Ind. Acad. Sci. A44 247 (1956)
  63. Pancharatnam S Proc. Ind. Acad. Sci. A44 398 (1956)
  64. Berry M V Proc. R. Soc. London A 392 45 (1984)
  65. Andronova I A, Malykin G B Usp. Fiz. Nauk 172 849 (2002); Andronova I A, Malykin G B Phys. Usp. 45 793 (2002)
© 1918–2024 Uspekhi Fizicheskikh Nauk
Email: ufn@ufn.ru Editorial office contacts About the journal Terms and conditions