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Canonical transformations from Jacobi to Whittaker

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Abstract

The idea of a canonical transformation emerged in 1837 in the course of Carl Jacobi's researches in analytical dynamics. To understand Jacobi's moment of discovery it is necessary to examine some background, especially the work of Joseph Lagrange and Siméon Poisson on the variation of arbitrary constants as well as some of the dynamical discoveries of William Rowan Hamilton. Significant figures following Jacobi in the middle of the century were Adolphe Desboves and William Donkin, while the delayed posthumous publication in 1866 of Jacobi's full dynamical corpus was a critical event. François Tisserand's doctoral dissertation of 1868 was devoted primarily to lunar and planetary theory but placed Hamilton–Jacobi mathematical methods at the forefront of the investigation. Henri Poincaré's writings on celestial mechanics in the period 1890–1910 succeeded in making canonical transformations a fundamental part of the dynamical theory. Poincaré offered a mathematical vision of the subject that differed from Jacobi's and would become influential in subsequent research. Two prominent researchers around 1900 were Carl Charlier and Edmund Whittaker, and their books included chapters devoted explicitly to transformation theory. In the first three decades of the twentieth century Hamilton–Jacobi theory in general and canonical transformations in particular would be embraced by a range of researchers in astronomy, physics and mathematics.

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Notes

  1. The idea of Hamilton-Jacobi theory as a distinct subject area seems to have taken hold in the last few decades of the nineteenth century. Schering (1873) published “Hamilton-Jacobische Theorie für Kräfte” and von Weber (1900) in a substantial article on partial differential equations for the Encyklopädie included a section on “Die Hamilton-Jacobi'sche Theorie”. In the calculus of variations one also finds “Hamilton-Jacobi theory” as a subject area: see Bolza (1909, 595–601), Bliss (1946, Chapter III) and Courant (1953, II 1–35). (Concerning the term “theory” Demidov (1982, 325–326) writes: “Mathematicians use the word theory in two essentially different meanings. In a narrow sense, it denotes a complex structure based on definite ideas and methods and covering a certain range of studies (thus, the theory of Galois, or Lagrange's theory of first-order partial differential equations). In a broad sense, the word theory designates a province of thought (e.g., theory of numbers; of differential equations.” In the case of Hamilton-Jacobi theory we are evidently using the first meaning of the word “theory.”).

  2. Nordheim and Fues’ tripartite division of Hamilton–Jacobi theory was also adopted in such standard mid-century textbooks as Lanczos (1949) and Corben and Stehle (1950). What distinguished Goldstein's perspective was the notion that the term Hamilton–Jacobi theory properly only applied to the third part of this division.

  3. The account which follows draws on Lützen (1990, 638–642) and Cayley (1858, 3–9).

  4. For the related researches of Lagrange and Poisson on the variation of constants at this time see Costabel (1981, 484–485).

  5. The first word of the title is spelled “Ueber” as in German spelling at the time. In the 1890 edition in Jacobi's Gesammelte Werke 5 the word is spelled “Über”. We have adopted the original spelling in our study.

  6. Weierstrass/Kotter writes “it would be highly unusual for Jacobi to have left such an important essay unpublished, unless he believed that it needed to be revised in some places.” (Jacobi 1890, Werke 5, 514).

  7. In the foreword to the Vorlesungen (1866a) it is stated that there is some uncertainty concerning the dating of the five supplementary works. The first four are said to be probably (“wahrscheinlich”) composed sometime after the Königsberg lectures of 1842-1843. However, no evidence is given for this judgement. Helmut Pulte (Jacobi (1996, xxxviii n. 110)) dates Ueber diejenigen Probleme to 1836 or into 1837, although it is possible the manuscript may have been completed somewhat later than this, although definitely before 1842.

  8. There is a misprint in the third equation of the first line, where \(\frac{{\text{d}}U}{{\text{d}}t}\) should be \(\frac{{\text{d}}U}{{\text{d}}z}\).

  9. English translation by Cajori (1929, 236).

  10. There is the possibility that Borchardt or Clebsch added the two symbols (d and ) for regular and partial differentiation in the 1866 transcription of Jacobi’s Königsberg lectures, where only one symbol may have been used. There is no evidence that this was the case. Indeed, the available evidence points to the opposite conclusion. The two symbols are used in Jacobi’s (1837a) as we saw above. Both symbols appear as well as in Jacobi (1846), written in 1838. Jacobi (1841) publicly advocated for using both symbols. Jacobi’s (1837c) famous article on the calculus of variations, “Zur Theorie der Variationsrechnung und der Differentialgleichungen,” uses the cursive for both ordinary and partial differentiation. The two symbols appear in all of Jacobi’s German-language publications on mathematical dynamics. Furthermore, in Pulte’s 1996 edition of the 1847-1848 Berlin lectures he includes some copies from the original handwritten notes taken by Scheibner. Consider in particular p. LX where the two symbols for ordinary and partial differentiation are employed; the corresponding typeset text is on p. 188.

  11. Cajori (1929, 235) writes of the “notation of partial derivatives which has become so popular in recent years.” Even at this late date the dual notation d/∂ was regarded as fairly novel.

  12. The title of the paper, “On the reduction of the integration of partial differential equations of the first order between any number of variables to the integration of a single system of ordinary differential equations,” is puzzling, since the integration theorem at the centre of the paper is the converse of what is stated in the title. The title better describes Cauchy's (1819) paper on first-order partial differential equations.

  13. In Ueber diejenigen Probleme Jacobi (1866b, 355) refers to this theorem as a generalization of Hamilton’s theorem. Courant and Hilbert (1962, 127) call Jacobi’s integration theorem the Hamilton–Jacobi theorem, although it is generally called Jacobi’s theorem in modern textbooks.

  14. See Nakane and Fraser (2002, 184–185).

  15. In discussing fundamental laws of mechanics, Truesdell (1968, 242 n. 4) calls attention to the utility of variational principles: “formal rearrangements are possible. E.g., a sufficiently general “variational” principle (i.e., a formal expression in variations, not a true minimum principle) can be made equivalent [to these laws]”.

  16. Hamilton’s derivation is given by Nakane and Fraser (2002, 181). For this derivation in modern textbooks see Goldstein (1950, 217), Fox (1954, 144-145), Gelfand and Fomin (1963, 67-70), Landau and Lifshitz (1969, 131-132) or Arthurs (1975, 19-20).

  17. Jacobi implicitly assumed that for each point t, qi, one and only one extremal connects the initial time and position to t, qi. In modern calculus of variations q = qi(t) is said to be embedded in a field of extremals containing the initial and final values of t, qi. The function \(\int \varphi dt\) in Eq. (19) is known as a field integral.

  18. In the Crelle article Jacobi (1837a) did not use canonical coordinates and Eqs. (29) are derived in the form.

    $$\frac{\partial S}{\partial {x}_{i}}={m}_{i}{\dot{x}}_{i}, \frac{\partial S}{\partial {y}_{i}}={m}_{i}{\dot{y}}_{i}, \frac{\partial S}{\partial {z}_{i}}={m}_{i}{\dot{z}}_{i} \left(i=1,\ldots ,\mu \right) ,$$

    where S is the principal function that is denoted by V in the Vorlesungen.

  19. Jacobi writes sinϕ2 for sin2ϕ.

  20. The term “Newtonian” here refers in the usual way to force equations given in terms of second derivatives of the coordinate variables.

  21. There is a fair degree of variation in the notation adopted by researchers for Lagrange and Poisson brackets. The appendix at the end of this article is a table giving the notation used by each researcher.

  22. The term “generating function” had become standard by the 1960s. The following are the designations for generating function adopted by the researches considered in this study: Jacobi (1866a) wilkürliche Function; Desboves (1848) function quelconque; Donkin (1855) modulus of a normal transformation; Poincaré (1892) function quelconque; Charlier (1907) Transformationsfunction; Klein (1926) Leitfunktion; Born (1925); Nordheim and Fues (1927) and Carathéodory (1935), Erzeugende Funktion der Transformation; Goldstein (1950) generating function; Corben and Stehle (1950) generator; Gelfand and Fomin (1963) generating function.

  23. The quoted passage is open to other objections. Jacobi developed a proof for Theorem X that was independent of Theorem IX because in point of fact Theorem IX does not imply Theorem X. In Sect. 42 he showed that for conservative systems Theorem X implies Theorem IX but the converse does not seem to be true.

  24. Felix Klein (1926, 203 and 1969, 191-192) seems to suggest that the transformation theorem (Theorem X of Ueber diejenigen Probleme) was deduced by Jacobi from his integration theorem (Theorem VI of Ueber diejenigen Probleme). This was not the case, and Jacobi himself regarded the transformation theorem as a new result. Furthermore, whereas the integration theorem was presented by Jacobi as a result in general dynamical theory, the transformation theorem was ostensibly introduced within the theory of variation of arbitrary constants. (Klein is intent on asserting Hamilton's importance (long overlooked in his view) and seems even to attribute the idea of a canonical transformation to the Irishman. On Klein’s “discovery” of Hamilton see Hankins (1980, 203–204).).

  25. The notation in Sect. 41 of Ueber diejenigen Probleme and lecture 21 of the Vorlesungen differ. The symbols ψ, W, α in the Vorlesungen become respectively H,—ψ, -α in Ueber diejenigen Probleme.

  26. Demidov (1982, 340) comments on the delay in publication of Jacobi's work and observes “Some of his achievements became known by word of mouth through his former students at Königsberg (Κ. W. Borchardt and others) …” However, no specific documentation is provided by Demidov for this assertion.

  27. Desboves evidently took for granted that the reader would understand that n is equal to 3 m, where m is the number of particles.

  28. Desboves denotes the solution to the H-J Eq. (117) as \(\theta \) where Jacobi calls it V, and Desboves' arbitrary constant C is called h by Jacobi. Also there is a minor variation in how they indicate the range of the subscripts.

  29. In (116) we set \({b}_{i}^{*}=-{b}_{i},\) so that (116) becomes \(\frac{{\text{d}}{b}_{i}^{*}}{{\text{d}}t}=-\frac{{\text{d}}(-{\Omega })}{\partial {a}_{1}},\quad \frac{{\text{d}}{b}_{i}^{*}}{{\text{d}}t}=\frac{{\text{d}}(-{\Omega })}{\partial {a}_{2}}\). The relations (124) given in terms of the generating function \(\theta \) become \( \frac{{\text{d}}\theta }{\text{d}{\alpha }_{i}}={\beta }_{i}, \frac{{\text{d}}\theta }{{\text{d}}{a}_{1}}= - {b}_{i}^{*}\). By the transformation theorem it then follows that (119) hold.

  30. Cayley's (1858, 26) remarks on Desboves’ memoir leave the impression that the two theorems proved by Desboves in the first part of his memoir are applied in the second part, which is not true. The theorem on canonical elements is only mentioned by Desboves at the end as a basis for further study of these elements. The transformation theorem itself had only been used to prove the theorem on canonical elements and did not come up again.

  31. Referring to English mathematicians of note, Hamilton in 1851 mentioned Herschel, Cayley, Donkin, Peacock and De Morgan (Crilly 2006, 178). Peacock and De Morgan were formalists and Cayley's mathematical orientation was algebraic and formalistic. Donkin himself published a paper in 1850 titled “On Certain Theorems in the Calculus of Operations,” research which was informed by Boole's formal operational approach to analysis.

  32. Jacobi’s integration theorem appears in Donkin (1854, 87–99); Jacobi’s theorem on canonical elements in Donkin (1854, 105–106); and Jacobi’s transformation theorem in Donkin (1855, 315–316). Concerning Jacobi’s theorem on canonical elements, Donkin (1854, 106 note) stated that the result was given by Jacobi (no specific citation is provided) and proved by Desboves (1848) but that his own proof was different from Desboves’. In his proof Donkin did not (as Desboves had) use a canonical transformation with a generating function that is a solution to a partial differential equation. Instead, Donkin’s proof uses Poisson bracket methods.

  33. The reasoning is verified in simple examples. Suppose for instance that n = 1 and we have the two pairs of variables x,y and a,b. Let \(X={a}^{2}{x}^{2}\). Equations (131) give y(x,a) =  2\(x{a}^{2}\) and b(x,a) = 2 \(a{x}^{2}.\) Then we find that a(x,y) = \(\sqrt{\frac{y}{2x}}\) and b(x,y) =  \(\sqrt{2{x}^{3}y}\). The identity

    $$\frac{\partial b(x,a)}{\partial a}\bullet \frac{\partial a\left(x,y\right)}{\partial y}=\frac{\partial b(x,y)}{\partial y}$$

    then becomes

    $$2{x}^{2}\bullet \frac{1}{\sqrt{8xy}}=\sqrt{\frac{{x}^{3}}{2y}},$$

    and the left and right sides of this equation are indeed equal. (We have here used the modern symbol for the partial derivative.).

  34. One might fault Donkin for not having given some details on the statement of this result in Jacobi (1837b). A reader coming to his paper would not be aware that the ideas of normal transformations and functional moduli originated with Jacobi. (His reference to Desboves’ paper on two theorems of Jacobi (1837b) did not include its title or much detail about its contents.).

  35. It is important to note that the term “canonical” is used just once in Jacobi (1837b) and that this is the only place that this term does appear in his published work prior to 1866. Donkin himself uses the term to characterize the symmetric form of Hamilton’s equations. The word appears thirteen times in Donkin (1855) (but not at all in Donkin (1854)).

  36. On British symbolic algebra in the first part of the nineteenth century see Pycior (1981). On the emergence of invariant theory in British algebra in the 1840s see Wolfson (2008).

  37. Carl Borchardt was in possession of notes for Jacobi’s Königsberg 1842–43 lectures, notes that were published in edited form as Jacobi (1866a). Wilhelm Scheibner had taken notes of Jacobi's 1847–48 Berlin lectures; his transcript would eventually be edited and published by Helmut Pulte as Jacobi (1996). Of course, Jacobi's contributions to canonical transformations were contained not in these works but in the set of notes that were published as Jacobi (1866b), but were probably written sometime around 1840. However, these notes do not appear to have undergone any detailed posthumous editing and apparently did not circulate. See also note 26 above.

    There are not copious biographical sources on Donkin. We do know that he displayed an early talent for languages (O’Connor and Robertson, 2021). It is worth noting that the term “canonical” does not appear at all in Donkin (1854), but is an established term in Donkin (1855). It would certainly be possible that there was some contact in (say) 1855 between Donkin and German mathematicians who had attended Jacobi’s lectures or had some familiarity with his unpublished writings. But there is no record of such contact in print or apparently in any of the biographical sources we have.

  38. In 1873 in an address to the Royal Astronomical Society Cayley noted that “everything in the Lunar theory is laborious” (Crilly 2006, 240).

  39. The meaning of these variables in terms of the lunar orbital parameters is as follows. Let a be the semi-major axis, e be the eccentricity, i be the inclination of the orbit to a fixed plane, μ be the sum of the masses of the earth and moon, l be the mean anomaly, g be the angular distance of the lower apsis from the ascending node, h be the longitude of the ascending node, then \(L=\sqrt{\mu a}\), \(G=L\sqrt{1-{e}^{2}}\), \(H=Gcosi\).

  40. A limitation of Delaunay’s theory was the slow convergence of his series. The standard lunar ephemerides published in Britain and United States in the 1920s were based on the different lunar theory of Hill and Ernest W. Brown (see Wilson (2010)). Delaunay's methods have, however, enjoyed a new lease on life with the advent since the 1950s of computer-assisted computation in applied celestial mechanics (artificial satellites, planetary perturbations), a development that is reviewed by Cook (1988, Chapter 9).

  41. The committee for Tisserand’s doctoral examination was Delaunay as president with Joseph Serret and Charles Briot as examiners. The first thesis was the work published in Liouville’s journal; the second thesis he defended was on astronomical refraction and did not apparently issue as a publication.

  42. Leo Koenigsberger (1904) (then in his mid-60 s) wrote a very mathematical “life and letters” biography of Jacobi but no major biographical study has appeared since then. Also, there are modern biographies for secondary figures (who did not work on canonical transformations) such as Hamilton and Cayley. See Hankins (1980) for Hamilton and Crilly (2006). There are no biographies older or modern for Desboves, Donkin, Tisserand, Charlier and Whittaker.

  43. English translations of passages in the Méthodes nouvelles are from the 1967 NASA English edition (Poincaré 1967). This translation is also used with some emendations in the American Institute of Physics edition of volume one (Poincaré 1993).

  44. Poincaré (1890) introduced such transformations on p.119, pp.170–171 and p.175; English translation (Poincaré 2017), p.104, pp.151–152 and p.155. For the history of this memoir see Barrow Green (1997).

  45. In modern dynamics the conjugate yi to xi is defined as the partial derivative of the Lagrangian L with respect to the time derivative of xi: \({y}_{i}=\frac{\partial L}{\partial {\dot{x}}_{i}}\), where \({\dot{x}}_{i}=\frac{{\text{d}}x}{{\text{d}}t}.\) Poincaré’s terminology is consistent with this definition.

  46. Poincaré switches the upper range of the variables from p to q, an apparent misprint.

  47. The 1866 volume in which the Vorlesungen as well as Ueber diejenigen Probleme appeared was itself titled Vorlesungen über Dynamik. It is possible that Poincaré for this reason referred to the source of Jacobi’s transformation theorem in this way. See also the discussion at the end of §6.5.3.

  48. The chapter in which the problem appears in Tisserand (1889) is Chapter VII, “Intégration des équations différentielles du mouvment elliptique par la méthode de Jacobi”. While the method is attributed to Jacobi the solution itself is Tisserand’s own invention.

  49. According to Brown (1896a, b, 1) the Gaussian constant of attraction is defined as follows. For any two masses m and m’ a distance r apart the gravitational force between them is \( F = k {\frac{mm^{\prime}}{{r}^{2}}} \). k is the Gaussian constant of attraction. The units may be chosen so that k = 1.

  50. In terms of the Delaunay elements identified in note 39, Poincaré is setting µ = 1.

  51. It should be noted that the symbol “'∂” as it was understood by Poincaré is not quite the partial derivative in the modern sense. Assume we have the function f(x,y) and the function g(x,\(x{^{\prime}}\)), where \({x}{^{\prime}}=\frac{{\text{d}}x}{{\text{d}}t}\) is the time derivative of x. Poincaré wrote the partial derivative of f with respect to x as \(\frac{{\text{d}}f}{{\text{d}}x}\) and the partial derivative of g with respect to x as \(\frac{\partial g}{\partial x}\). The “∂ rond” is used when x occurs in an expression that also contains x’ but the “d” is used otherwise.

  52. In his excelllent introduction to the revised English translation of Volume One of Les méthodes nouvelles (that also extends to later aspects of Poincaré’s work), Daniel Goroff (1993, I26-27) presents Poincaré’s 1899 derivation of Jacobi’s theorem on canonical transformations. However, Goroff does not point out that the proof is seldom if ever attributed to Poincaré in modern textbooks.

  53. For a modern account of one of Tisserand's (1896) results in his investigation of the three-body problem see Murray and Dermott (2012, 71–73).

  54. For example, Eq. (6.7) from Méthodes Nouvelles becomes Eq. (6.45) in the Leçons.

  55. (6.50) evidently expresses the fact that the generating function S is a function of y and y’ and that the following relations hold.

    $$\frac{\partial S}{\partial y}=-x, \frac{\partial S}{\partial {y}{^{\prime}}}={x}{^{\prime}}.$$
  56. Modern textbooks standardly derive Jacobi’s transformation theorem using the 1899 proof. However, an exception is ter Haar (1964, 99–100), where the derivation follows the ideas of Poincaré (1905), although ter Haar's account is less rigorous and complete than Poincaré's. Another exception is Brouwer and Clemence (1961, 531–533) whose proof bears some similarities to those of Jacobi (1866b) and Charlier (1907).

  57. Courant and Hilbert (1962, 107–109, 129–131) give both proofs of Jacobi’s integration theorem. Gelfand and Fomin (1963, 91–93) also give both proofs.

  58. Charlier did not give any references to the literature.

  59. The expressions for the Delaunay elements in terms of the elliptical orbital parameters were given in note 39 above. Charlier provided a similar set of expressions for the new element set; we will not follow him in his subsequent detailed investigation of perturbed planetary motion. This particular canonical transformation of the Delaunay elements appears in modern textbooks, for example Brouwer and Clemence (1961, 539–540).

    .

  60. It is noteworthy that references by authors from around 1900 to Jacobi's Vorlesungen and Ueber diejenigen Probleme are to the editions of these works published in Jacobi's Gesammelte Werke in 1886 and 1890 respectively, rather than to the original volume published in 1866. It suggests that the publication of the collected works may have led in the 1890s to a renewed interest in Jacobi's work.

  61. There are some points of similarity between Charlier’s work and the memoirs from the 1850s of Donkin – in choice of notation, the extension of generating functions to time-dependent functions, the use of Poisson square brackets and the division of generating functions into four types. Donkin's memoirs had appeared in the Philosophical Transactions, the premier English-language scientific journal in the nineteenth century. However, Charlier did not mention Donkin and may not have read his papers. Charlier’s proof of the theorem on canonical transformations followed Jacobi rather than Donkin.

  62. Charlier does not give the expressions (7.37) and (7.38) for \({\psi }_{3}, {\psi }_{2}\), \({\psi }_{4}\). They were given by Donkin (1855) and can be found in modern textbooks; see for example, Goldstein (1950, 240–243).

  63. In Corben and Stehle’s Classical Mechanics (1950) Sect. 68 the term contact transformation is used. In the index of the book under the entry “Canonical transformations” one finds “see Contact transformations.”.

    In Sect. 67 (p. 218) of Corben and Stehle's book a transformation is said to be a contact transformation from q,p to Q,P if there exists a function S = S(q,Q) such that pdq-PdQ = dS. The authors then give Poincaré's proof that such a transformation preserves the canonical form of the canonical equations (with no reference to Poincaré.) In Sect. 68 (p. 221) the authors write, “The essential reason for the introduction of contact transformations is the property exhibited by (67.3) [the canonical Eqs. (8.1) in our account] that they leave the form of the canonical equations invariant.”.

    In Goldstein’s Classical Mechanics (1950) the term “canonical transformation” is used in the same way. The terms contact transformation and canonical transformation mean the same thing in the two books. “Canonical transformation” has become the standard in today’s mechanics.

    (It should be noted that Carathéodory (1935, 78-121) some years earlier wrote in detail about contact and canonical transformations and the relation between them. He proved a theorem (Satz 1) in Sect. 121 on pp. 107-108 that asserts that the two concepts—general homogeneous canonical transformation and general contact transformation—are the same (except for matters of notation).).

  64. Whittaker (1904) doesn't identify where in Lie's work the relevant mathematical theory was developed. Presumably Lie and Georg Scheffers' Geometrie der Berührungstransformationen (Leipzig, 1896) was an important source. In the second (1917) edition Whittaker adds a new section at the beginning of Chapter XI where he uses ideas from Hamilton's optics to introduce transformations in a geometric way, and he here refers to Lie's contact transformation, again with no references to sources.

  65. Goldstein erred here, the relevant Chapters in Whittaker were X, XI and XII.

  66. For an informative account of perturbation theory in atomic physics in the 1920s see Fues (1927), in which the subject receives its own chapter in the Handbuch der Physik. Fues includes some remarks on the relation of perturbations in physics to traditional celestial mechanics. He writes (p. 132), “In the past, physics had little reason to be interested in this mode of calculation, until the establishment of Bohr's atomic model suddenly created a close relationship between atomic theory and cosmic astronomy.” For an historical study of this subject see Shore (2003).

  67. Vinit’s book on orbital and celestial mechanics contains a chapter on Hamilton–Jacobi theory that features Jacobi’s theorem on canonical elements. See Vinti (1998, Chapter7).

  68. In his biography of Jacobi, Koenigsberger (1904, 247) states Jacobi's transformation theorem in the part dealing with Jacobi's researches in Konigsberg at the end of the 1830s. He doesn't indicate where in Jacobi's writings the theorem is to be found or give many details about it but does comment that Jacobi applied it to the problem of attraction to two fixed centres. This problem is taken up in lecture 29 of the 1866 Vorlesungen where it is solved using Jacobi's integration theorem that had been presented in lecture 20. (Koenigsberger also doesn't give any of this information. The problem of attraction to two fixed centres is cited by Nakane and Fraser (2002, 215).) The transformation theorem itself does not appear in lecture 29 or anywhere else in the Vorlesungen. Koenigsberger does not appear to be aware that the transformation theorem was first stated in Jacobi's Paris note of 1837 and proved as Theorem X in the 1866 supplemental work Ueber diejenigen Probleme. (See also the discussion in Sect. 3.7.2 above.).

  69. It should be noted that despite Jacobi's strong algorithmic and formal tendencies, his work possessed a more developed sense of mathematical deduction and proof than was the case for such older figures as Lagrange. Jacobi's thinking about the foundations of mechanics and mathematics in relation to Lagrange is examined by Pulte (1996, 1997, 1998). Pulte draws attention to Jacobi’s Berlin lectures (Jacobi 1996), which were not published during the period although they appear (as lectures themselves or as lecture notes) to have influenced Bernhard Riemann and Carl Neumann. In terms of the work considered in the present study the most important influence on Jacobi was Hamilton rather than Lagrange.

  70. Jacobi was a remarkable figure and a leading mathematician of the nineteenth century; in Klein’s (1926) history of nineteenth-century mathematics he is accorded the same prominence as Gauss, Riemann and Weierstrass. However, Jacobi does not seem to enjoy a correspondingly high profile in today's mathematical culture. For example, he does not appear in Victor Katz's History of Mathematics, a widely read textbook today.

  71. A lack of a systematic geometric perspective affected not only Jacobi's contributions to mathematical dynamics but also his work in pure analysis. Demidov (1982, 339–340) states that this limitation prevented Jacobi from creating a general theory of first-order partial differential equations, something that would be achieved by Sophus Lie in the 1870s.

  72. English translation in Bell (1986, 544–545).

  73. English translation in Sommerfeld (1923, 555–556).

  74. Nordheim was Hilbert's assistant from 1922–1927, although his primary mentor was Born (Nordheim completed his physics PhD under Born in 1923). At this time Hilbert was entering retirement-age years while Nordheim was in his early to mid-20 s. The more mathematical character of Nordheim's writings in the 1920s compared to other physicists was no doubt influenced by his contact with Hilbert. In Hilbert (2009) the editors state that Nordheim and fellow Born student Gustav Heckmann “worked out” Hilbert's lectures on quantum mechanics from 1922–23 and Nordheim “worked out” Hilbert’s lectures on the same subject in 1926–1927. It is not clear precisely what “worked out” means in this context but it may simply have been that Nordheim and Heckmann wrote the lectures and Hilbert provided editorial emendations. Fifty years later Nordheim would reflect on his time with Hilbert in less than positive terms. In a 1977 interview with historian of physics Bruce Wheaton, he recounted: “First I must say, during the time I was his assistant, he [Hilbert] was very sick. And not the genius he had been. He lived very much in the past in a way. His mathematical interest was logic, which was not terribly appealing to me. But he had the conviction that the best thing for a young man was to work with him. That was a reward in itself. And everything else, financial and family considerations, would be way down in importance.” (From the website of the American Institute of Physics: https://www.aip.org/history-programs/niels-bohr-library/oral-histories/5074.) Of course, personal memories of events long in the past may be selective. From a mathematical viewpoint there are some interesting foundational aspects to Nordheim and Fues' 1927 article that were likely influenced by Nordheim's association with Hilbert (Nordheim was the first author on this article).

  75. Physicists Herbert Corben and Philips Stehle (1950, v) write, “The parts of classical mechanics which are of present-day interest to the physicist are not those which were of paramount interest in the nineteenth century. At present fundamental physics is the physics of particles and fields, whereas nineteenth-century mechanics was the study of the n-body problem. There are many points in common between these problems, but the points of view are vastly different.” See also Fues’ comment in note 66 above.

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Acknowledgements

The authors would like to thank Jesper Lützen and Helmut Pulte for reading and advising on the present paper. We are also grateful for the feedback that we have received at academic conferences (Asian History of Astronomy Group, Physical Society of Japan, American Mathematical Society, Canadian Mathematical Society and Canadian Society for the History and Philosophy of Mathematics) at various stages in the preparation of the present study. Research for this article was financially supported by the Social Sciences and Humanities Research Council of Canada, Japanese Grants-in-Aid for Scientific Research (JP17500686), and the Eleanor P. May Funds of the University of Toronto.

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Appendix: Notation for Lagrange and Poisson Brackets

Appendix: Notation for Lagrange and Poisson Brackets

We have two sets of variables qi, pj and ai, bj (i, j = 1, 2, …, n). The two sets of variables are functionally connected.

The Lagrange bracket as originally introduced by Lagrange (1809a, b) is defined as

$$ [{a_i},{b_j}] = \mathop \sum \limits_{k = 1}^n \left( {\frac{{\partial {q_k}}}{{\partial {a_i}}}\frac{{\partial {p_k}}}{{\partial {b_j}}} - \frac{{\partial {q_k}}}{{\partial {b_j}}}\frac{{\partial {p_k}}}{{\partial {a_i}}}} \right).$$

The Poisson bracket as originally introduced by Poisson (1809a, b) is defined as

$$ \left( {a_{i} ,b_{j} } \right) = \mathop \sum \limits_{k = 1}^{n} \left( {\frac{{\partial a_{i} }}{{\partial q_{k} }}\frac{{\partial b_{j} }}{{\partial p_{k} }} - \frac{{\partial a_{i} }}{{\partial p_{k} }}\frac{{\partial b_{j} }}{{\partial a_{i} }}} \right). $$

Some later authors adopted square brackets for Lagrange brackets and parentheses for Poisson brackets. Other authors adopted the reverse. The table below gives the convention followed by each author.

 

Lagrange bracket

Poisson bracket

Lagrange (1809a, 1811a, b)

Square [xx]

Round (xx)

Poisson (1809a, b)

Square [xx]

Round (xx)

Hamilton (1835)

Brace {xx}

Desboves (1848)

Square [xx]

Donkin (1854)

Square [xx]

Cayley (1858)(Cayley used the term “Coefficient” rather than “bracket” and employed the same round parentheses to denote both Lagrange and Poisson brackets. In the edition of the report that was published in Cayley's collected papers (1890) the following sentence is added in Sect. 13: “It may be noticed that throughout the Report, I speak of the Lagrange's Coefficients (a, b), and Poisson's Coefficients (a, b), distinguishing them in this manner, and not by any difference of notation.”.)

Round (xx)

Round (xx)

Jacobi (1866a)

Round (xx)

Square [xx]

Poincaré (1905)

Round (xx)

Square [xx]

Brown (1903)

Round (xx) and square [xx]

Charlier (1902)

Round (xx)

Square [xx]

Whittaker (1904)

Square [xx]

Round (xx)

Carathéodory (1935)

Square [xx]

Round (xx)

Nordheim and Fues (1927)

Square [xx]

Round (xx)

Carathéodory (1935)

Square [xx]

Round (xx)

Lanczos (1949)

Square [xx]

Round (xx)

Goldstein (1950)

Brace {xx}

Square [xx]

Corben and Stehle (1950)

Square [xx]

Round (xx)

Brouwer and Clemence (1961)

Square [xx]

Landau and Lifshitz (1969)

Square [xx]

Akhiezer (1988)

Square [xx]

Vinti (1998)

Square [xx]

Round (xx)

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Fraser, C., Nakane, M. Canonical transformations from Jacobi to Whittaker. Arch. Hist. Exact Sci. 77, 241–343 (2023). https://doi.org/10.1007/s00407-022-00303-9

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