Fedorov, Yuri; Pantazi, Chara The Picard-Fuchs equations for complete hyperelliptic integrals of even order curves, and the actions of the generalized Neumann system. (English) Zbl 1298.14037 J. Math. Phys. 55, No. 3, 032703, 8 p. (2014). The authors find the Picard-Fuchs type equations for the case of the family of even order genus 2 curves \[ \Gamma_h=\{ \omega^2=(z-a_1)(z-a_2)(z-a_3)(z^3+h_1z+h_2)\} \] which appear in quadratures of an integrable generalization of the Neumann system with a separable quartic potential. Reviewer: Bujar Fejzullahu (Presevo) MSC: 14H70 Relationships between algebraic curves and integrable systems 34M03 Linear ordinary differential equations and systems in the complex domain 34M56 Isomonodromic deformations for ordinary differential equations in the complex domain 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests Keywords:Neumann system; action variables; abelian integrals; Picard-Fuchs equation PDFBibTeX XMLCite \textit{Y. Fedorov} and \textit{C. Pantazi}, J. Math. Phys. 55, No. 3, 032703, 8 p. (2014; Zbl 1298.14037) Full Text: DOI arXiv References: [1] Abenda, S.; Fedorov, Y., On the weak Kowalevski-Painlevé property for hyperelliptically separable systems, Acta Appl. Math., 60, 2, 137-178 (2000) · Zbl 0984.37068 [2] Davison, C. M.; Dullin, H. R.; Bolsinov, A. V., Geodesics on the ellipsoid and monodromy, J. Geom. Phys., 57, 12, 2437-2454 (2007) · Zbl 1148.53026 [3] Dullin, H. R.; Richter, P. H.; Veselov, A. P., Action variables of the Kovalevskaya top, J. Moser at 70 (Russian). Regul. Chaotic Dyn., 3, 3, 18-31 (1998) · Zbl 0952.37044 [4] Dullin, H.; Richter, P. H.; Veselov, A. P.; Waalkens, H., Actions of the Neumann systems via Picard-Fuchs equations, Phys. D, 155, 3-4, 159-183 (2001) · Zbl 1001.70013 [5] Eilbeck, J.; Enolski, V.; Kuznetzov, V.; Tsiganov, A., Linear R-matrix algebra for classical separable systems, J. Phys. A: Math. Gen., 27, 567-578 (1994) · Zbl 0824.70008 [6] Enolski, V.; Richter, P., Periods of hyperelliptic integrals expressed in terms of θ-constants by means of Thomae formulae, Philos. Trans. R. Soc., A, 366, 1867, 1005-1024 (2008) · Zbl 1153.33308 [7] Ince, E. L., Ordinary Differential Equations (1965) · JFM 53.0399.07 [8] Lawden, D. F., Elliptic Functions and Applications, 80 (1989) · Zbl 0689.33001 [9] More precisely, the original Picard-Fuchs equations are second order equations obtained by elimination of the periods of the meromorphic differentials. [10] Moser, J., Various aspects of integrable Hamiltonian systems, Proceedings of the CIME Conference, Bressanone, Italy, 1978 [11] Moser, J., Geometry of Quadrics and Spectral Theory, 147-188 (1980) [12] Mumford, D., Tata lectures on theta II, Prog. Math., 43, 272 (1984) [13] Neumann, C., De probleme quodam mechanico, quod ad primam integralium ultra-ellipticoram classem revocatum, J. Reine Angew. Math., 56, 46-63 (1859) · ERAM 056.1472cj [14] Novikov, D.; Yakovenko, S., Redundant Picard-Fuchs system for abelian integrals, J. Differ. Eqs., 177, 2, 267-306 (2001) · Zbl 1011.37042 [15] Rauch-Wojciechowski, S.; Tsiganov, A. V., Integrable one-particle potentials related to the Neumann system and the Jacobi problem of geodesic motion on an ellipsoid, Phys. Lett. A, 107, 3, 106-111 (1985) · Zbl 1177.37060 [16] Schlesinger, L., Handbuch der Theorie der linearen Differentialgleichungen (1968) · JFM 26.0329.01 [17] This integral is slightly different from the canonical integral E(k), for this reason we use the notation \documentclass[12pt]{minimal}\( \begin{document}\bar{E}(k)\end{document} \). [18] Vanhaecke, P., Stratification of hyperelliptic Jacobians and the Sato Grassmannian, Acta Appl. Math., 40, 143-172 (1995) · Zbl 0827.14015 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.