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 Tr. Mat. Inst. Steklova, 2016, Volume 295, Pages 261–291 (Mi tm3761)

Abel's theorem and Bäcklund transformations for the Hamilton–Jacobi equations

A. V. Tsiganov

Saint Petersburg State University, Universitetskaya nab. 7-9, St. Petersburg, 199034 Russia

Abstract: We consider an algorithm for constructing auto-Bäcklund transformations for finite-dimensional Hamiltonian systems whose integration reduces to the inversion of the Abel map. In this case, using equations of motion, one can construct Abel differential equations and identify the sought Bäcklund transformation with the well-known equivalence relation between the roots of the Abel polynomial. As examples, we construct Bäcklund transformations for the Lagrange top, Kowalevski top, and Goryachev–Chaplygin top, which are related to hyperelliptic curves of genera 1 and 2, as well as for the Goryachev and Dullin–Matveev systems, which are related to trigonal curves in the plane.

 Funding Agency Grant Number Russian Science Foundation 15-11-30007 This work is supported by the Russian Science Foundation under grant 15-11-30007.

DOI: https://doi.org/10.1134/S0371968516040166

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English version:
Proceedings of the Steklov Institute of Mathematics, 2016, 295, 243–273

Bibliographic databases:

UDC: 517.958+512.77

Citation: A. V. Tsiganov, “Abel's theorem and Bäcklund transformations for the Hamilton–Jacobi equations”, Modern problems of mechanics, Collected papers, Tr. Mat. Inst. Steklova, 295, MAIK Nauka/Interperiodica, Moscow, 2016, 261–291; Proc. Steklov Inst. Math., 295 (2016), 243–273

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tm3761
• https://doi.org/10.1134/S0371968516040166
• http://mi.mathnet.ru/eng/tm/v295/p261

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This publication is cited in the following articles:
1. Andrey V. Tsiganov, “Bäcklund Transformations for the Nonholonomic Veselova System”, Regul. Chaotic Dyn., 22:2 (2017), 163–179
2. A. V. Tsiganov, “Bäcklund transformations for the Jacobi system on an ellipsoid”, Theoret. and Math. Phys., 192:3 (2017), 1350–1364
3. A. V. Tsiganov, “Duffing Oscillator and Elliptic Curve Cryptography”, Nelineinaya dinam., 14:2 (2018), 235–241
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