Abstract
We propose a method to construct first integrals of a dynamical system, starting with a given set of linearly independent infinitesimal symmetries. In the case of two infinitesimal symmetries, a rank two Poisson structure on the ambient space it is found, such that the vector field that generates the dynamical system, becomes a Poisson vector field. Moreover, the symplectic leaves and the Casimir functions of the associated Poisson manifold are characterized. Explicit conditions that guarantee Hamilton–Poisson realizations of the dynamical system are also given.
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References
Abraham R., Marsden J.E.: Fondations of Mechanics. Benjamin Cummings, New York (1978)
Arnold, V.I.: Mathematical Methods of Classical Mechanics. 2nd edn. Graduate Texts in Mathematics, vol. 60 Springer, New York (1989)
Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical aspects of classical and celestial mechanics, Itogi Nauki i Tekhniki, Sovr. Probl. Mat. Fundamentalnye Napravleniya, vol. 3, VINITI, Moscow 1985. English transl.: Encyclopedia of Math. Sciences, vol. 3, Springer, Berlin (1989)
Dufour, J.-P., Zung, N.T.: Poisson Structures and their Normal Forms. Birkhäuser, New York (2005)
Kozlov, V.V.: Remarks on a Lie theorem on the integrability of differential equations in closed form, Differential Equations, 41(4), 588–590 (2005). Translated from Differentsial’nye Uravneniya, 41(4), 553–555 (2005)
Kozlov, V.V.: Simmetriya, topologiya i rezonansy v gamil’tonovoi mekhanike (Symmetry, topology, and resonances in Hamiltonian mechanics), Izhevsk, (1995)
LaSalle J.P.: The extent of asymptotic stability. Proc. Acad. Sci. USA 46, 363–365 (1960)
Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry, 2nd edn., Texts in Applied Mathematics, vol. 17. Springer, Berlin (1999)
Moser, J., Zehnder, E.: Notes on Dynamical Systems, Courant Lecture Notes in Mathematics, vol. 12, American Mathematical Society (2005)
Nakanishi N.: Poisson cohomology of plane quadratic Poisson structures. Publ. RIMS Kyoto Univ. 33, 73–89 (1997)
Ratiu, T.S., Tudoran, R.M., Sbano, L., Sousa Dias, E., Terra, G.: Geometric Mechanics and Symmetry: the Peyresq Lectures; Chapter II: A Crash Course in Geometric Mechanics, London Mathematical Society Lecture Notes Series, vol. 306, pp. 23–156. Cambridge University Press, Cambridge (2005)
Tudoran R.M.: A normal form of completely integrable systems. J. Geom. Phys. 62(5), 1167–1174 (2012)
Vaisman, I.: Lectures on the Geometry of Poisson Manifolds. Birkhäuser, New York (1994)
Verhulst, F.: Nonlinear Differential Equations and Dynamical Systems. 2nd edn., Springer, New York (2006)
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Tudoran, R.M. A Method to Generate First Integrals from Infinitesimal Symmetries. Mediterr. J. Math. 13, 249–262 (2016). https://doi.org/10.1007/s00009-014-0470-6
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DOI: https://doi.org/10.1007/s00009-014-0470-6