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 TMF, 1998, Volume 115, Number 1, Pages 3–28 (Mi tmf854)

Homogeneous Stäckel-type systems

A. V. Tsiganov

St. Petersburg State University, Faculty of Physics

Abstract: A class of Hamiltonian dynamic systems integrated by the variable separation method is considered. The integration for this class is the inversion of an Abel mapping on hyperelliptic curves. We prove that the derivative of the Abel mapping is the Stäckel matrix, which determines a diagonal Riemannian metric and curvilinear orthogonal coordinate systems in a flat space. Lax representations with the spectral parameter are constructed. The corresponding classical $r$-matrices are dynamic. It is shown how the class of pointwise canonical transformations can be naturally generalized using the Abel integral reduction theory.

DOI: https://doi.org/10.4213/tmf854

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English version:
Theoretical and Mathematical Physics, 1998, 115:1, 377–395

Bibliographic databases:

Citation: A. V. Tsiganov, “Homogeneous Stäckel-type systems”, TMF, 115:1 (1998), 3–28; Theoret. and Math. Phys., 115:1 (1998), 377–395

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. V. Tsiganov, “Outer automorphisms of $sl(2)$, integrable systems, and mappings”, Theoret. and Math. Phys., 118:2 (1999), 164–172
2. A. V. Tsiganov, “Noncanonical time transformations relating finite-dimensional integrable systems”, Theoret. and Math. Phys., 120:1 (1999), 840–861
3. Tsiganov, AV, “Automorphisms of sl(2) and classical integrable systems”, Physics Letters A, 251:6 (1999), 354
4. A. V. Tsiganov, “Canonical transformations of the extended phase space and integrable systems”, Theoret. and Math. Phys., 124:1 (2000), 918–937
5. A. V. Tsiganov, “Degenerate integrable systems on the plane with a cubic integral of motion”, Theoret. and Math. Phys., 124:3 (2000), 1217–1233
6. A. V. Tsiganov, “Construction of Separation Variables for Finite-Dimensional Integrable Systems”, Theoret. and Math. Phys., 128:2 (2001), 1007–1024
7. Klishevich, VV, “Exact solution of Dirac and Klein-Gordon-Fock equations in a curved space admitting a second Dirac operator”, Classical and Quantum Gravity, 18:17 (2001), 3735
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