The original impetus for Sophus Lie's work was apparently to streamline the solution of certain problems of variational type such as those treated in the work of Euler and Lagrange. This presumably involves solution of differential equations involving groups of symmetries. Is there a reference that explains this historical background?
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asked Jan 29, 2018 at 15:24
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2$\begingroup$ I've never gone back to Lie's original work, but Armand Borel says a little about it in his historical study mathscinet.ams.org/mathscinet-getitem?mr=1847105 (while the link takes you to his review of a more detailed book by Tom Hawkins on the history of Lie theory; Chapter 1 in Hawkins may be more helpful than Borel's short account). My rough impression has always been that Lie set out to imitate Galois theory, for differential equations rather than polynomials. The Ritt-Kolchin theory of diffeential algebraic groups inherits some of this program: see Kolchin's book. $\endgroup$– Jim HumphreysJan 31, 2018 at 19:01
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3$\begingroup$ @JimHumphreys: just for historical background, the "set out to imitate Galois theory [...]" at 2018-01-31 19:01:34Z brough to mind a caustic yet very relevant comment in a proposal to appoint Friedrich Schottky as successor of Lazarus Fuchs at Berlin. The proposal was written by a committee (the undersigned are G. Frobenius, H. A. Schwarz, J. Bauschinger, W. Foerster, Carl Stumpf, M. Planck), and it is reproduced in [K.-R. Biermann: Die Mathematiker und ihre Dozenten an der Berliner Universität 1810-1933, p. 317]. The comment I am speaking of is: "Originell ist das Problem, die [...] $\endgroup$– Peter HeinigFeb 6, 2018 at 16:49
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3$\begingroup$ [...] continuierlichen Gruppen zu untersuchen, wenig originell aber die Ausführung, die mechanische Übertragung der feinen Begriffe, welche die genialsten Algebraiker in der Theorie der endlichen Gruppen geschaffen hatten, trivial sind die Resultate, nahezu bedeutungslos die Anwendungen, worin gezeigt wird, wie ein auf natürlichem Wege von Euler oder Lagrange erledigtes Problem mittelst der von Lie erdachten Methoden sehr viel umständlicher gelöst werden kann." [my translation:] The problem to investigate the continuous groups is original, the execution of this task by way of mechanically... $\endgroup$– Peter HeinigFeb 6, 2018 at 16:54
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3$\begingroup$ ...imitating the subtle notions created by the most ingenious algebraists in the theory of finite groups, is rather unoriginal; the results are trivial, the applications---in which it is showg how to solve a problem already solved by Euler or Lagrange along a natural route in a much more cumbersome way via the methods invented by Lie---are almost irrelevant. Frobenius--Schwarz et al. are unambiguously speaking of Lie here (I left out the more personal attacks), and they are apparently writing these, strictly speaking, irrelevancies, in order to eliminate two competitors... $\endgroup$– Peter HeinigFeb 6, 2018 at 17:00
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3$\begingroup$ ...for a position that they want Schottky to get: Friedrich Engel and Friedrich Schur, who, according to the committee, both work in the manner of Lie, and, so they argue, because of Lie's unimportance, should both be discarded for Berlin. The "imitating the subtle notions created by the most ingenious algebraists in the theory of finite groups" presumably means the same as "to imitate Galois theory" in your comment. My comments are not to endorse this curious damnation of Lie, signed by no less than four famous names (Frobenius, Schwarz, Stumpf, Planck), just to add context. $\endgroup$– Peter HeinigFeb 6, 2018 at 17:07
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1 Answer
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In
Vladimir Itskov: Orbit Reduction of Contact Ideals. Contemporary Mathematics. Vol. 285, 2001
one reads on page 172:
As first observed by Sophus Lie [8], the Euler-Lagrange equations of every invariant variational problem can be written in terms of the differential invariants of the group action. In other words, the Euler-Lagrange equations of a group-invariant variational problem can be pushed forward to the orbit space. Surprisingly, up to date there was no general understanding of the meaning of the pushed forward equations on the orbit space, nor there was a general algorithm of producing the group-invariant Euler-Lagrange equations.
In the above, the "nor there was a" seems not quite correct English and should be 'nor was there a'; the reference '[8]' is:
Sophus Lie Über Integralinvarianten und ihre Verwertung für die Theorie der Differentialgleichungen, Leipziger Berichte 49, 369-410, 1897.
and a slightly commented/edited/expanded/corrected version of the above seems to be available here; (I don't know what the referent of the initials 'G.S.' in that document is.)
answered Jan 29, 2018 at 21:28
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$\begingroup$ The editors are Alf Guldberg and Carl Størmer, and the initials 'G.', 'S.', and 'G. S.' appear in various places, presumably indicating which of the editors has contributed a relevant translation or gloss. $\endgroup$– LSpiceSep 7, 2018 at 19:47