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On the Integration of the Canonic Equations and the Principle of Duality

Published online by Cambridge University Press:  20 January 2009

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The object of this paper is to essay an analytical statement of the reduction of the integration of a canonic system of differential equations (into which time does not enter explicitly) to that of the partial differential equation of Jacobi and Hamilton; and to illustrate the principle of duality by an outline of the solution for the problem of two bodies both by the standard form of the equation referred to and by the analogous form which that principle involves. Most statements of the reduction are verifications and somewhat obscure the symmetry of the canonic form. The shortest procedure, of course, is by means of the well known theorem of Jacobi, and this verificatory method is followed by Tisserand, Charlier and Appell. Poincaré gives a proof depending on a simple form given by him to the conditions for a canonical change of variables, but again the statement lacks analytical form. The essentials of this proof will be given here, but in an entirely different way. An analytical treatment of the subject has been given by Professor L. Becker in his class lectures at Glasgow, but it has not been published.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1909