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On the role of virtual work in Levi-Civita’s parallel transport

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Abstract

The current literature on history of science reports that Levi-Civita’s parallel transport was motivated by his attempt to provide the covariant derivative of the absolute differential calculus with a geometrical interpretation (For instance, see Scholz in The intersection of history and mathematics, Birkhäuser, Basel, pp 203–230, 1994, Sect. 4). Levi-Civita’s memoir on the subject was explicitly aimed at simplifying the geometrical computation of the curvature of a Riemannian manifold. In the present paper, we wish to point out the possible role implicitly played by the principle of virtual work in Levi-Civita’s conceptual reasoning to formulate parallel transport.

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Notes

  1. Biographical notes are taken from Levi-Civita’s obituary by Ugo Amaldi (1875–1957), Levi-Civita (1954), vol. 1, IX–XXX.

  2. Veronese spent a period of time studying in Leipzig under the supervision of Felix Klein (1849–1825). He provided contributions to projective hyper-spaces and non-Euclidean geometry.

  3. Flores d’Arcais graduated in Pisa, and had Enrico Betti (1823–1892) and Ulisse Dini (1845–1918) among his teachers. He is well known in the Italian school of mathematics of the days for his excellent handbooks on calculus.

  4. Ricci-Curbastro graduated in Pisa having Betti, Dini, and Eugenio Beltrami (1835–1900) among his teachers. He perfected his studies with Klein and, jointly with Levi-Civita, is considered the father of absolute differential calculus.

  5. A pupil of Beltrami, Padova investigated mathematical physics in non-Euclidean spaces.

  6. It is well known that Albert Einstein (1879–1955) claimed to feel indebted with him for absolute calculus.

  7. Parallel transport and linear (affine) connection were introduced almost simultaneously by Levi-Civita, Gerhard Hessenberg (1874–1925), Hermann Weyl (1885–1955) and Jan Arnoldus Schouten (1883–1971) in connection with Einstein’s general theory of relativity. Levi-Civita used (local) embedding of a Riemannian manifold in some n-dimensional space. Weyl introduced parallel transport (hence, linear connection) on arbitrary differential manifolds on a completely general basis, that is, with no reference to Riemann metrics, see Maurin (1997), ch. 1, Sect. 1.1.

  8. Sect. 1.3, p. 214.

  9. Sect. 3, p. 337.

  10. Levi-Civita (1917), Introduction.

  11. According to Lagrange Lagrange (1853), in the period 1736–1742 the Bernoullis, Alexis Clairaut (1713-1765) and Leonhard Euler (1707–1783) were among the first to assimilate constraint reactions to active forces.

  12. In Lagrange (1853), 1st ed., p. 179; our translation. See also Capecchi Capecchi (2012), p. 15.

  13. Lagrange (1853), 1st ed., p. 11; our translation.

  14. Lagrange (1853), 3rd ed., p. 21.

  15. Lagrange claims such a thesis is D’Alembert’s, but D’Alembert’s actual principle is different Capecchi (2012).

  16. Lagrange (1853), 1st ed., p. 14.

  17. See Levi-Civita and Amaldi (1949), vol. 1, ch. XV; vol. 2, part 1, ch. V, Sect. 3, n. 18–21; Levi-Civita and Amaldi (1965), part 1, ch. XIV, Sect. 2 , n. 4–8; part 2, ch. V, Sect. 3, n. 17–19; Agostinelli and Pignedoli (1961), vol. 2, ch. V, Sect. 1, n. 4; Agostinelli and Pignedoli (1988), ch. I, Sects. 1, 2; Finzi (1968), vol. 1, ch. XIII, Sect. 4.

  18. Also said to be D’Alembert-Lagrange principle as reformulated by Lagrange, Arnold (1986), ch. IV, or general equation of virtual work, Belluzzi (1961), vol. I, ch. XV, Sect. 318. See also the references in the previous footnote.

  19. See also Pizzocchero (1998), Sect. 3.

  20. Levi-Civita (1917), p. 3.

  21. In doing so, Levi-Civita was the first to deal with forms of pseudo-Riemannian structures, see Bottazzini (1990), pp. 305–306.

  22. Bianchi (1924b), ch. XXV.

  23. Levi-Civita (1917), Eq. (1), p. 4.

  24. Levi-Civita (1917), Eq. (4), p. 5.

  25. Levi-Civita (1917), Eq. (7), p. 6.

  26. Levi-Civita (1917), p. 7.

  27. Levi-Civita (1917), Eq. (I), p. 7.

  28. Grassini (1988), ch. 3, Sect. 2 , n. 2.6.

  29. Levi-Civita (1917), unnumbered equation before Eq. (8), p. 7.

  30. Levi-Civita (1917), Eq. (8), p. 7.

  31. Levi-Civita (1917), Eq. (I\(_a\)), p. 8.

  32. Levi-Civita (1954), vol. IV, p. 8, Bianchi (1924a), ch. II.

  33. Levi-Civita used the term ‘moment’, which was traditional in the Italian school of mathematical physics of his time and denoted a mechanical action dual to a Lagrangian parameter of admissible (virtual) displacements.

  34. Levi-Civita (1917), eq. (I\(_c\)), p. 12. The same comment on Eq. (14) may be found in later textbooks of the Italian school of mathematical physics, e.g. Finzi and Pastori (1960), ch. X.

  35. Levi-Civita (1917), Sect. 13.

  36. Bernardini (1974), ch. XII, Krall (1940), part 1, ch. IV, Sect. 3.

  37. Sommerfeld (1957), ch. II, Sect. 8.

  38. See Levi-Civita (1924), pp. 97–143.

  39. Levi-Civita (1924), Sect. 2, p. 99.

  40. Levi-Civita (1924), Eq. (1), p. 102. Note that in the Italian school of mathematical physics of the time the symbol for the scalar product was \(\times \).

  41. Levi-Civita (1924), Eq. (1\('\)), p. 104.

  42. Levi-Civita (1927), ch. V, (b), Sects. 10–15. This monograph was suggested by the mathematician Edmund Taylor Whittaker (1873–1956), who wanted to widen the first Italian edition of 1925 with the addition of a third part, devoted to the physical application of absolute differential calculus to Einstein’s general relativity, extracted from Levi-Civita (1928).

  43. We could not find any biographical data on him so far, apart from the fact that he published some notes on intrinsic parallelism.

  44. One of Levi-Civita’s pupils, who edited his lectures on the absolute differential calculus, Levi-Civita (1927).

  45. Levi-Civita (1927), pp. 102–104 (pp. 119–121 of the Italian edition).

  46. Levi-Civita (1927), ch. XI, Sect. 12. See Davies and Yano (1975) for a historical account of the influence of Levi-Civita’s parallelism on differential geometry.

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  1. University of Palermo, Palermo, Italy

    Giuseppe Iurato

  2. Sapienza University, Rome, Italy

    Giuseppe Ruta

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  1. Giuseppe Iurato
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Communicated by: Umberto Bottazzini.

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Iurato, G., Ruta, G. On the role of virtual work in Levi-Civita’s parallel transport. Arch. Hist. Exact Sci. 70, 553–565 (2016). https://doi.org/10.1007/s00407-016-0177-0

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