Abstract
This chapter is entirely devoted to Lagrange’s VWL. In the first part the first introduction of the law of Lagrange is reported, which has a wording similar to that of Bernoulli. Lagrange calls his VWL and Bernoulli’s the principle of virtual velocities. In the central part the wordings of VWL in the two editions of the Mécanique analytique in terms of virtual displacements (following a foundational route) and the Théorie des fonctions analytique in terms of virtual velocity (following a reductionist route) are presented. In the final part an overview of D’Alembert’s mechanics is presented aimed at an understanding of the extensibility of VWLs to dynamics.
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Notes
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p. 108.
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Preface, translation by I. Bruce.
- 3.
p. 107.
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pp. 216–218.
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p. 319.
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p. 12.
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p. 179.
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p. 2.
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pp. 1–2.
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p. 17.
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p. 240.
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pp. 245–246.
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pp. 8–9.
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In the transcription I made a slight typographical variation of Lagrange’s formulas, from the beginning and then even in relation (A), instead to indicate the element of mass with α, I used the symbol dm, putting it after the force per unit mass rather than before. The same notation was used in Lagrange’s Théorie de la libration de la Lune.
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The negative signs are due for the first three terms to the principle of D’Alembert, for which the accelerating forces must be treated with sign changed, for the last two terms, to the convention on solar and terrestrial gravity forces which are considered positive if attractive, while the change of distance is positive if there is an increment of distance.
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p. 9.
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p. 10.
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p. 197.
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pp. 15–16.
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p. 16.
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p. 20.
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pp. 44–45.
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pp. 10–11.
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pp. 11–12.
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p. 12.
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p. 137.
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pp. 15–16.
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p. 16.
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p.16.
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p. 28. The equations of moments are deduced on pp. 28–29.
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pp. 101–115.
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pp. 45–46.
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p. 49.
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p. 49.
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p. 23.
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p. 137.
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p. 26.
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pp. 255–256.
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p. 384.
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pp. 384–385.
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p. 366.
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p. 12.
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p. 181.
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pp. 17–18.
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p. 255.
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p. 256.
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p. 73.
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p. XVII.
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p. XII.
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p. 26.
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pp. 3,4, 35, 50–51.
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pp. 73–75.
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pp. 335–337.
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p. 233.
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pp. 72–73, part II.
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p. 73, part II.
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pp. 96–97.
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Capecchi, D. (2012). Lagrange’s contribution. In: History of Virtual Work Laws. Science Networks. Historical Studies, vol 42. Springer, Milano. https://doi.org/10.1007/978-88-470-2056-6_10
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