Abstract
We construct a symplectic realization and a bi-Hamiltonian formulation of a 3-dimensional system whose solution are the Jacobi elliptic functions. We generalize this system and the related Poisson brackets to higher dimensions. These more general systems are parametrized by lines in projective space. For these rank 2 Poisson brackets the Jacobi identity is satisfied only when the Plücker relations hold. Two of these Poisson brackets are compatible if and only if the corresponding lines in projective space intersect. We present several examples of such systems.
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Adler, M., van Moerbeke, P., and Vanhaecke, P., Algebraic Integrability, Painlevé Geometry and Lie Algebras, Ergeb. Math. Grenzgeb. (3), vol. 47, Berlin: Springer, 2004.
Borisov, A.V. and Mamaev, I. S., Rigid Body Dynamics: Hamiltonian Methods, Integrability, Chaos, Izhevsk: R&C Dynamics, Institute of Computer Science, 2005 (Russian).
Borisov, A.V. and Mamaev, I. S., Isomorphisms of Geodesic Flows on Quadrics, Regul. Chaotic Dyn., 2009, vol. 14, nos. 4–5, pp. 455–465.
Braden, H.W., Gorsky, A., Odesskii, A., and Rubtsov, V., Double-Elliptic Dynamical Systems from Generalized Mukai–Sklyanin Algebras, Nuclear Phys. B, 2002, vol. 633, no. 3, pp. 414–442.
Braden, H.W., Marshakov, A., Mironov, A., and Morozov, A., On Double-Elliptic Integrable Systems: 1. A Duality Argument for the Case of SU(2), Nuclear Phys. B, 2000, vol. 573, nos. 1–2, pp. 553–572.
Laurent-Gengoux, C., Pichereau, A., and Vanhaecke, P., Poisson Structures, Grundlehren Math. Wiss., vol. 347, Heidelberg: Springer, 2013.
Cushman, R.H. and Bates, L.M., Global Aspects of Classical Integrable Systems, 2nd ed., Basel: Birkhäuser, 2015.
Damianou, P.A., Nonlinear Poisson Brackets, PhD Thesis, Univ. of Arizona, Tucson, 1989, 107 pp.
Damianou, P.A., Transverse Poisson Structures of Coadjoint Orbits, Bull. Sci. Math., 1996, vol. 120, no. 2, pp. 195–214.
Damianou, P.A., Sabourin, H., and Vanhaecke, P., Transverse Poisson Structures to Adjoint Orbits in Semisimple Lie Algebras, Pacific J. Math., 2007, vol. 232, no. 1, pp. 111–138.
Dragović, V. and Gajić, B., On the Cases of Kirchhoff and Chaplygin of the Kirchhoff Equations, Regul. Chaotic Dyn., 2012, vol. 17, no. 5, pp. 431–438.
Dubrovin, B.A., Theta-Functions and Nonlinear Equations, Russian Math. Surveys, 1981, vol. 36, no. 2, pp. 11–92; see also: Uspekhi Mat. Nauk, 1981, vol. 36, no. 2(218), pp. 11–80.
Fairlie, D.B., An Elegant Integrable System, Phys. Lett. A, 1987, vol. 119, no. 9, pp. 438–440.
Grabowski, J., Marmo, G., and Perelomov, A.M., Poisson Structures: Towards a Classification, Modern Phys. Lett. A, 1993, vol. 8, no. 18, pp. 1719–1733.
Joswig, M. and Theobald, Th., Polyhedral and Algebraic Methods in Computational Geometry, London: Springer, 2013.
Meyer, K. R., Jacobi Elliptic Functions from a Dynamical Systems Point of View, Amer. Math. Monthly, 2001, vol. 108, no. 8, pp. 729–737.
Nambu, Y., Generalized Hamiltonian Dynamics, Phys. Rev. D (3), 1973, vol. 7, pp. 2405–2412.
Odesskii, A.V. and Rubtsov, V.N., Polynomial Poisson Algebras with a Regular Structure of Symplectic Leaves, Theoret. and Math. Phys., 2002, vol. 133, no. 1, pp. 1321–1337; see also: Teoret. Mat. Fiz., 2002, vol. 133, no. 1, pp. 3–23.
Perelomov, A. M., Some Remarks on the Integrability of the Equations of Motion of a Rigid Body in an Ideal Fluid, Funct. Anal. Appl., 1981, vol. 15, no. 2, pp. 144–146; see also: Funktsional. Anal. i Prilozhen, 1981, vol. 15, no. 2, pp. 83–85.
Sklyanin, E.K., Some Algebraic Structures Connected with the Yang–Baxter Equation, Funct. Anal. Appl., 1982, vol. 16, no. 4, pp. 263–270; see also: Funktsional. Anal. i Prilozhen., 1982, vol. 16, no. 4, pp. 27–34, 96.
Takhtajan, L., On Foundation of the Generalized Nambu Mechanics, Comm. Math. Phys., 1994, vol. 160, no. 2, pp. 295–315.
Vanhaecke, P., Integrable Systems in the Realm of Algebraic Geometry, 2nd ed., Lect. Notes Math., vol. 1638, Berlin: Springer, 2001.
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Damianou, P.A. Poisson Brackets after Jacobi and Plücker. Regul. Chaot. Dyn. 23, 720–734 (2018). https://doi.org/10.1134/S1560354718060072
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DOI: https://doi.org/10.1134/S1560354718060072