PAMM * Proc. Appl. Math. Mech. 15, 705 – 706 (2015) / DOI 10.1002/pamm.201510342 The role of virtual work in Levi-Civita's parallel transport Giuseppe Iurato1 and Giuseppe Ruta2,∗ 1 Department of Physics and Chemistry, University of Palermo, Italy 2 Department of Structural and Geotechnical Engineering, University "La Sapienza", Rome, Italy According to current history of science, Levi-Civita introduced parallel transport solely to give a geometrical interpretation to the covariant derivative of absolute differential calculus. Levi-Civita, however, searched a simple computation of the curvature of a Riemannian manifold, basing on notions of the Italian school of mathematical physics of his time: holonomic constraints, virtual displacements and work, which so have a remarkable, if not dominant, role in the origin of parallel transport. c© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction Tullio Levi-Civita (1873-1941)'s teachers gave him a strong training in mathematical physics and mechanics, and developed absolute differential calculus, an inheritance of Beltrami's investigations on Riemannian manifolds [1, 2]. In the mid-1910s, Levi-Civita investigated relativity by absolute differential calculus: he studied the curvature of four-dimensional Riemannian manifolds, modelling space-time, through the parallel transport of vectors over these manifolds. We usually find that Levi-Civita's parallel transport is motivated by the search for a geometrical interpretation to the covariant derivative of absolute differential calculus. However, in the introduction to [3] Levi-Civita states that he wishes to simplify the computation of the curvature of a Riemannian manifold Vn with dimension n≥ 2, and introduces parallelism on it. Going further, the principle of virtual work seems to be one of the basis for parallel transport according to Levi-Civita. 2 Virtual work and parallel transport Balance exists if and only if the total power (work) on any admissible, or virtual, velocity (first-order displacement) field vanishes. This law is basically the same since Lagrange's formulation, where inertia is a force to be added to the active ones, and dynamics is reduced to statics.1 The reactions of perfect constraints spend no work on admissible displacements; if ~δPi is the first-order displacement of the point where active forces plus inertia ~Fi are applied, the law of virtual work is ([5], p. 14) δL = ∑ i ~Fi * ~δPi = 0, (1) Levi-Civita in [3] embeds Vn in a Euclidean space SN with dimension N ≤ n(n+ 1)/2, and considers two unit vectors ~α,~α ′ emerging from nearby points P,P′ ∈ Vn. In SN , ~α,~α ′ are parallel if, for any auxiliary direction ~f emerging from P, angle(~α, ~f ) = angle(~α ′, ~f ). Parallelism in Vn asks this relation to hold for any direction ~f of the tangent plane T SN P (Vn), which is an intrinsic definition, since it depends solely on metrics in Vn, ds2 = ∑ni,k=1 aikdxidxk. The manifold Vn may be described by the system ([3], eq. (1), p. 4) yν = yν(x1, . . . ,xn), ν = 1,2, . . . ,N (2) with yν ∈ SN , xn ∈Vn. Eq. (2) describes an n d.o.f. system subjected to N smooth holonomic bilateral constraints. The point P varies on a smooth curve C ∈Vn, parameterized by the abscissa s, thus the components αν = αν(s), so that C ≡ yν(s) = yν(x1(s), . . . ,xn(s)), ν = 1, . . . ,N ⇒ y′ν = n ∑ i=1 ∂yν ∂xi x′i ν = 1,2, ...,N, (3) In the analog constrained system, s is an evolution parameter (e.g., time), and C is a trajectory in the manifold of admissible configurations. Derivation with respect to s yields the direction cosines y′ν ∈ SN , while x′i are the direction cosines of the same unit direction in Vn. Levi-Civita poses the direction cosines of ~α to be ξ (i), i = 1,2, . . . ,n with respect to Vn, and αν , ν = 1, . . . ,N with respect to SN . Then, from eq. (3) we have ([3], eq. (7), p. 6) αν = n ∑ l=1 ∂yν ∂xl ξ (l) ν = 1,2, . . . ,N. (4) ∗ Corresponding author: e-mail giuseppe.ruta@uniroma1.it, phone +39 (0)6 4458 5085, fax +39 (0)6 4458 5292 1 Lagrange gives credit to D'Alembert, but the actual D'Alembert's principle is different [4]. c© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 706 Section 24: History of mechanics Levi-Civita considers an arbitrarily fixed direction ~f of SN , with direction cosines fν , thus cos ( ~f ,~α ) = N ∑ ν=1 αν fν ⇒ d cos ( ~f ,~α ) = ds N ∑ ν=1 α ′ ν(s) fν . (5) Levi-Civita's intrinsic condition of parallelism requires the variation in eq. (5) to vanish only for the directions tangent to Vn at P ∈ C , which are those compatible with the constraints (2) ([3], p. 7). Thus, by replacing the fν with quantities proportional to them, Levi-Civita's intrinsic definition of parallelism implies ([3], eq. (I), p. 7) N ∑ ν=1 α ′ ν(s)δyν = 0 (6) for any variation δyν , that is, for any admissible virtual displacement compatible with the constraints in eq. (2). If the α ′ν(s) are some mechanical actions in SN , eq. (6) is a formulation of the virtual work law in SN related to the smooth bilateral holonomic system defined by eq. (2), hence related to a Riemannian manifold. From eq. (2) it follows that ([3], p. 7) δyν = n ∑ k=1 ∂yν ∂xk δxk ν = 1,2, . . . ,N,⇒ N ∑ ν=1 α ′ ν(s) ∂yν ∂xk = 0 (k = 1,2, . . . ,n), (7) for the arbitrariness of the δxk and (6). Eq. (7) are the parallelism conditions for the directions ~α moving along C . To have an intrinsic expression, Levi-Civita replaces the direction cosines αν with their expression eq. (4), to include the intrinsic direction cosines ξ (i), and, if Γ jli are Christoffel symbols of second kind, he deduces ([3], eq. (Ia), p. 8) dξ (i) ds + n ∑ j,l=1 Γ jl i x ′ jξ (l) = 0 (i = 1,2, . . . ,n), (8) Levi-Civita shows that eq. (8) may be expressed by the covariant quantities associated with the ξ (i) 2, thus the intrinsic condition of parallelism has the same form of Lagrange equations of motion on a Riemannian manifold ([3], eq. (Ic), p. 12). It seems apparent that one key guide to Levi-Civita is mathematical physics, not only pure geometry, and, in particular, the principle of virtual work for a mechanical system under smooth holonomic bilateral constraints. 3 Final remarks We claim that the virtual work principle played a key role in the origin of Levi-Civita's parallel transport in a Riemannian manifold, eq. (6). Levi-Civita's jargon clearly refers to this, e.g., with the locutions 'constraint' and 'displacements compatible with constraint'. Furthermore, he specifies that eq. (6) is obtained "for all the displacements δyν compatible with the constraints (2)", highlighting in italic this phrase in [3]. Indeed, for the analog constrained mechanical system, the unit directions emerging from the points of Vn assume the role of admissible displacements, and the dual forms on them are reactions provided by the geometrical links between the elements of the mechanical system ([6], ch. II, sect. 8). In his monograph on absolute differential calculus [7] (ch. V, (b), sects. 10-15), Levi-Civita highlights the role of analytical mechanics in developing differential geometry notions, and says that his symbolic equation of parallelism formally recalls the principle of virtual work. He uses the same principle in discussing the geodetic principle for the dynamics of a material particle moving in a four-dimensional space-time manifold ([7], ch. XI, sect. 12). It seems apparent to us that the key notions of the Italian school of mathematical physics were a strong basis for LeviCivita's non-Euclidean geometrical definitions. References [1] D. Capecchi, G. Ruta, Strength of Materials and Theory of Elasticity in 19th Century Italy (Springer Int. Publ., Cham, 2015). [2] R. Tazzioli, Arch. Hist. Exact Sciences 46 1–37 (1993). [3] T. Levi-Civita, in: Opere matematiche. Memorie e note (6 vol, Zanichelli, Bologna, 1954-73), vol 4, pp. 1–39. [4] D. Capecchi, History of Virtual Work Laws (Birkhäuser, Boston, 2012). [5] J.L. Lagrange, Mécanique analytique (Desaint, Paris, 1788; Courcier, Paris, 1811-1815; Mallet-Bachelier, Paris, 1853). [6] A. Sommerfeld, Vorlesungen über Theoretische Physik. Band 1: Mechanik (Becker & Erler, Leipzig, 1943). [7] T. Levi-Civita, Lezioni di calcolo differenziale assoluto (A. Stock, Roma, 1925; English transl.: Blackie & Son, London, Glasgow, 1927). 2 Levi-Civita adopts the locution 'moment', traditional in the Italian school of mathematical physics of his time, defining a mechanical action dual to a Lagrangean parameter of admissible (virtual) displacements. c© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.gamm-proceedings.com