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On those principles of mechanics which depend upon processes of variation

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Literatur

  1. „Über das Prinzip der Aktion und über die Klasse mechanischer Prinzipien, der es angehört”, Math. Ann. Ann. Bd. 58, 1904, pp. 169–194; „Bemerkungen zur Note des Herrn Philip E. B. Jourdain über das Prinzip der kleinsten Aktion”, ibid., „Über das Prinzip der Aktion und über die Klasse mechanischer Prinzipien, der es angehört”, Math. Ann. Ann. Bd. 64, 1907, pp. 156–159.

  2. In this form, ‘the principle of least action’ has been given in most text-books since Jacobi's time; for example, Darboux, ‘Leçons sur la théorie générale des surfaces’, t. 2, Paris 1889, pp. 491–500; ‘ell, ‘Traité de mécanique rationnelle’, 2e ed., tei corpi, Milano, 1896, pp. 394–396.

  3. „Geschichte des Prinzips der kleinsten Aktion”, Akad. Antrittsvorlesung, Leipzig, 1877, p. 27–29.

  4. Math. Ann. Bd. 58, 1904, pp. 171–172.

  5. Cf. § 3 below.

  6. See Hölder, „Über die Prinzipien von Hamilton und Maupertuis”, Gött. Nachr., 1896, pp. 122–157 and § 3 below. Only Réthy's manner of expression leads to the confusion of his view with that of Ostrogradsky; in reality, his view is that or Rodrigues and Mayer (1886), and, under these limitations and extensions as to the equations of condition, of Hölder. Hölder's general priciple is\(\mathop \smallint \limits_{t_0 }^{t_2 } \left( {\delta T + 2T\frac{{d\delta t}}{{dt}} + \delta U} \right)dt = 0,\) Hamilton's principle results from this when the δ-process is further defined by the condition δt=0, and the principle of least rction results when the δ-process is defined, by the condition, δ′U = δTt is not zero). Thus these two last principles are fundamentally quite distinct. Further, it was indicated by Hölder that the transformation into general coordinates was to be carried out in the way developed in § 3 below.

  7. In his ‘Rigid Dynamics’ since 1877; cf. 6th ed. (1905) of Part II (‘Advanced Part’), pp. 301 sqq.

  8. Hölder was quite conscious of this fact (see a note in Quart. Journ. of Math., 1904, p. 75).

  9. C. Neumann (1888), Hertz (1894), Hölder (1896) und Appell (1898); see also Boltzmann, „Vorlesungen über die Prinzipe der Mechanik”, Teil II, Leipzig 1904, pp. 30–34.

  10. Quart. Journ. of Math., 1904, pp. 72, 75; Math. Ann. Bd. 62, 1906, pp. 415, 417–418. Réthy's remark (Math. Ann. Bd. 64, 1907, pp. 156–157) that my statement that\(\delta \mathop \smallint \limits_{t_0 }^{t_2 } 2T \cdot dt = 0\) is only true, without further discussion, if the conditions do not containt explicitly, is correct if δ is aliteral variation; not correct if (as I assumed) δ is a Hölder's ‘variation’. It would have been better not to write\(\delta \mathop \smallint \limits_{t_0 }^{t_2 } 2T \cdot dt = 0\), but to keep the form:\(\delta \mathop \smallint \limits_{t_0 }^{t_2 } (2T \cdot d\delta t + 2\delta _1 T \cdot dt) = 0\), whereδ 1 T is a ‘variation’ (not strictly speaking) defined by Hölder's process; but I followed Hölder's precedent, and also that in Encykl. der math. Wiss. IV 1, p. 93. My investigations (Quart. Journ. of Math., 1905, pp. 290–294) also use Hölder's δ-process, exclusively.

  11. Op. cit., Quart. Journ. of Math., 1904, art. 446, p, 303.

  12. Ibid. Quart. Journ. of Math., 1904, I controverted this remark in Quart. Journ. of Math., 1904, p. 75, and Math. Ann. Bd. 62, 1906, pp. 417–418, because I was under the impression that, with Routh, δx is defined by\(\sum\limits_v {\tfrac{{\partial x}}{{\partial q_v }}\delta q_v + \tfrac{{\partial x}}{{\partial t}}\delta t} \), theq ν's being generalised coordinates;\(\delta x - \frac{{\partial x}}{{\partial t}}\delta t\) is then the only expression for a virtual displacement. But see the above text, and introduction.

  13. This aspect of Réthy's work previously escaped me. I tacitly used Hölder's process of ‘variation’, while Réthy (strictly speaking, correctly) did not admit that it was a ‘variation’ at all (on this point, of which Hölder was perfectly conscious, see Quart. Journ. of Math, 1904, p. 75, note). Thus I wrongly attributed to Réthy, Routh and Voss certain errors. In fact, I assumed, and still assume, the greater importance of Hölder's method of forming a concept of a (not literal) variation which remains valid for non-holonomous systems; no literally variational process being capable of this.

  14. Math. Ann. Bd. 58, 1904, p. 173. I only know of his earlier (1895–1896) work from this paper. A somewhat more general formulation, brought about by addingc times the dentity δ(Φ·dt)−d(Φ·St)≡δ′Φ·dt,c being an arbitrary constant, to (7) was given by Réthy in ibid., Bd. 64, 1907, pp. 157–158.

  15. Voss'theorem cannot be inverted (see Math. Ann. Bd. 58, 1904, pp. 174–175), and the above invertible theorem was given in ibid., pp. 175–176 and Bd. 64, 1907, pp. 157–158.

  16. „Über die principe von Hamilton und Maupertuis” [July, 1900], Gött. Nachr., Math.-Phys. Klasse, 1900, pp. 322–327. We need not here consider the special case first (§ 1) considered by Voss, that the conditions do not depend ont explicitly; but will at once proceed to the general case.

  17. Loc. cit., „Über die Principe von Hamilton und Maupertuis” [July, 1900], Gött. Nachr., Math.-Phys. Klasse, 1900, p. 14, note.

  18. If all then q ν's weremutually independent, thenany system of variations given to them would result in a virtual displacement of the mechanical system and any real variation of aq ν iseo ipso virtual, as remarked in the note on p. 417 of the Math. Ann. Bd. 62. Réthy wrongly attributed (ibid., Math. Ann. Bd. 64, p. 158) to me the remark that\(\delta q_v - \dot q_v \cdot \delta t\) is not, in general, virtual: what I said (ibid., Math. Ann. Bd. 62, pp. 418–419) was that, if δx is any variation (affectingt also, and defined by\(\delta x = \sum\limits_v {\left( {\tfrac{{\partial x}}{{\partial q_v }}\delta q_v + \tfrac{{\partial x}}{{\partial t}}\delta t} \right)} \) ofx, δx−x·δt is not virtual,—for\(\delta x - \frac{{\partial x}}{{\partial q_v }}\delta t\) is. The question as to whether a certain displacement of a coordinate is ‘virtual’ or not can only arise when the coordinate is not independent. Here the general case is considered: theq's fix the position of the system at the timet—and, indeed, over-determine it, for further equations of condition, which need not be integrable, hold between thedq's.

  19. Denoted δ1 T by me in Math. Ann. Bd. 62, 1906, p. 416.

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Siehe die Arbeiten der Herren Jourdain und Réthy in den Bänden 62 und 64 dieser Annalen. Da sich auch Herr Réthy mit den neuen Ausführungen Herrn Jourdains einverstanden erklärt hat, so schließen wir die Diskussion.

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Jourdain, P.E.B. On those principles of mechanics which depend upon processes of variation. Math. Ann. 65, 513–527 (1908). https://doi.org/10.1007/BF01451167

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