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On those principles of mechanics which depend upon processes of variation. (English) JFM 38.0695.02

Die Bemerkungen von Réthy (JFM 38.0695.01) betreffen die Note von Jourdain, über welche F. d. M. 37, 722, 1906, JFM 37.0722.01 referiert ist; einzelne Schlüsse dieser Note werden angezweifelt. Von der ausführlichen Erwiderung Jourdains führen wir die allgemeinen Gedanken der Einleitung wörtlich an.
“Unter den Fragen von mathematischem Interesse und den Streitpunkten betreffs der Variationsprinzipien der Mechanik sind die vornehmsten die bezüglich der exakten Bedeutung des benutzten Prozesses der Variation, nämlich ob diese betrachtet werden kann als eine buchstäbliche (eigentliche) Variation oder nicht. Diese Fragen betreffen: (a) das Verhältnis des Prinzips der kleinsten Aktion zum Hamiltonschen Prinzip und in Verbindung damit die Frage, ob die Variable \(t\) in dem erstern zu variieren ist (in dem letzteren ist \(\delta t=0\)); (b) die Ausdehnung der Prinzipien, – ob und unter welchen Bedingungen sie anwendbar sind auf die Fälle der Nichtexistenz einer Kräftefunktion, nichtholonomer Bedingungen, solcher Bedingungen, welche explizit von \(t\) abhängen –, und die damit zusammenhängende Schwierigkeit in bezug auf die Transformation der Prinzipien aus rechtwinkligen in allgemeine koordinaten. Seit der Zeit von Lagrange sind beide Fragen in einer mehr oder weniger expliziten Form erstanden, so in den Arbeiten von Rodrigues, Jacobi, Ostrogladsky, Routh, A. Mayer, Sludsky, Bertrand, Voss, Helmholtz, Réthy, Hölder, Appel und bei mir selbst. Den historischen Teil beabsichtige ich anderswo zu erledigen. Hier will ich unter besonderer Bezugnahme auf einige neuere Abhandlungen von Réthy solche Antworten auf beide Fragen geben, die mir befriedigend zu sein scheinen.” Die subtilen Auseinandersetzungen müssen im Orginale nachgelesen werden. In einer Fußnote erklärt die Redaktion der Mathematischen Annalen: “Da sich auch Herr Réthy mit den neuen Ausführungen Herrn Jourdains einverstanden erklärt hat, so schließt wir die Diskussion”.

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[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. · JFM 34.0760.05
[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 ?-process is further defined by the condition ?t=0, and the principle of least rction results when the ?-process is defined, by the condition, ??U = ?T (?t 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 ? is aliteral variation; not correct if (as I assumed) ? 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: \(\delta \mathop \smallint \limits_{t_0 }^{t_2 } (2T \cdot d\delta t + 2\delta _1 T \cdot dt) = 0\) , where? 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 ?-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, ?x is defined by \(\sum\limits_v {\tfrac{{\partial x}}{{\partial q_v }}\delta q_v + \tfrac{{\partial x}}{{\partial t}}\delta t} \) , theq ?’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 ?(?{\(\cdot\)}dt)?d(?{\(\cdot\)}St)????{\(\cdot\)}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 ?’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 ? 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 ?x 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, ?x?x{\(\cdot\)}?t 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 ?1 T by me in Math. Ann. Bd. 62, 1906, p. 416.
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