# American Institute of Mathematical Sciences

September  2018, 10(3): 293-329. doi: 10.3934/jgm.2018011

## The Euler-Poisson equations: An elementary approach to integrability conditions

 1 Institute of Metal Science, Equipment and Technologies, with Hydro- and Aerodynamics Centre "Acad. A. Balevski", Bulgarian Academy of Sciences, 67 Shipchenski Prohod Street, 1574 Sofia, Bulgaria 2 Université Paris 13, Sorbonne Paris Cité, CNRS UMR 7539, Laboratoire Analyse, Geometrie et Aplications, 99 Av. J.-B. Clement, 93430 Villetaneuse, France

* Corresponding author

Received  April 2017 Revised  August 2018 Published  August 2018

We consider the Euler-Poisson equations describing the motion of a heavy rigid body about a fixed point with parameters in a complex domain. We suppose that these equations admit a first integral functionally independent of the three already known integrals which does not depend on all the variables. We prove that this may happen only in the already known three integrable cases or in the trivial case of kinetic symmetry. We provide a method for finding such a fourth integral, when it exists.

Citation: Sasho Popov, Jean-Marie Strelcyn. The Euler-Poisson equations: An elementary approach to integrability conditions. Journal of Geometric Mechanics, 2018, 10 (3) : 293-329. doi: 10.3934/jgm.2018011
##### References:
 [1] M. Adler and P. van Moerbeke, The algebraic integrability of geodesic flow on SO(4), Invent. Math., 67 (1982), 297-331.  doi: 10.1007/BF01393820.  Google Scholar [2] Yu. A. Arkhangelskii, Analytical Dynamics of the Rigid Body, Nauka, Moscow, 1977. (in Russian).  Google Scholar [3] Yu. A. Arkhangelskii, On one new property of Euler-Poisson equations, Doklady Akad. Nauk USSR, 258 (1981), 810-811. (in Russian).  Google Scholar [4] V. I. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Math., 60, 2$^{nd}$ edition, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar [5] M. Audin, Spinning Tops. A Course on Integrable Systems, Cambridge Studies in Advanced Mathematics, 51, Cambridge University Press, 1996.  Google Scholar [6] M. Audin, Les Systèmes Hamiltoniens et Leur Intégrabilité, Cours spécialisé, 8, Société Mathématique de France, EDP Sciences, 2001, (French) [Hamiltonian Systems and Their Integrability], SMF/AMS Texts and Monographs, 15, American Mathematical Society, 2008.  Google Scholar [7] O. I. Bogoyavlenskii, Euler equations on finite-dimensional Lie coalgebras, arising in problems of mathematical physics, Uspekhi Mat. Nauk, 47 (1992), 107-46, (Russian) [Euler equations on finite-dimensional Lie coalgebras, arising in problems of mathematical physics], Russian Math. Surveys, 47 (1992), 117-189. doi: 10.1070/RM1992v047n01ABEH000863.  Google Scholar [8] O. I. Bogoyavlenskii, Integrable Euler equations on Lie algebras arising in problems of mathematical physics, Izv. Akad. Nauk SSSR Ser. Mat., 48 (1984), 883-938, (Russian) [Integrable Euler equations on Lie algebras arising in problems of mathematical physics], Math. USSR Izvestiya, 25 (1985), 207-257. doi: 10.1070/IM1985v025n02ABEH001278.  Google Scholar [9] O. I. Bogoyavlenskii, Integrable Euler equations on six-dimensional Lie algebras, Doklady Akad. Nauk USSR, 268 (1983), 11-15, (Russian) [Integrable Euler equations on sixdimensional Lie algebras], Soviet Math. Doklady, 27 (1983), 1-5.  Google Scholar [10] A. V. Borisov and I. S. Mamaev, Poisson Structures and Lie Algebras in Hamiltonian Mechanics, Library "R & C Dynamics", Edited by Izdatel'skiĭ Dom "Udmurtskiĭ Universitet", Izhevsk, 1999. (in Russian). http://ics.org.ru/publications/index.php?cat=103&author=23  Google Scholar [11] A. V. Borisov and I. S. Mamaev, Rigid Body Dynamics. Hamiltonian Methods, Integrability, Chaos, Moscow-Izhevsk: Institute of Computer Science, 2005. http://ics.org.ru/publications/index.php?cat=103&author=23 Google Scholar [12] A. V. Borisov and I. S. Mamaev, Modern Methods of the Theory of Integrable Systems, Moscow-Izhevsk: Institute of Computer Science, 2003. http://ics.org.ru/publications/index.php?cat=103&author=23  Google Scholar [13] A. T. Fomenko, Integrability and Nonintegrability in Geometry and Mechanics, Dodrecht: Kluwer Academic Publishers, 1988. doi: 10.1007/978-94-009-3069-8.  Google Scholar [14] I. N. Ganshenko, G. V. Gorr and A. M. Kovalev, Classical Problems of Rigid Body Dynamics, Kiev, Naukova Dumka, 2012. (in Russian). Google Scholar [15] V. V. Golubev, Lectures on Integration of the Equations of Motion of a Rigid Body about a Fixed Point, Moscow, Gostekhizdat, 1953, (Russian) [Lectures on Integration of the Equations of Motion of a Rigid Body about a Fixed Point], Israel program for scientific translations, Haifa, 1960.  Google Scholar [16] B. Grammaticos, J. Moulin-Ollagnier, A. Ramani, J.-M. Strelcyn and S. Wojciechowski, Integrals of quadratic ordinary differential equations in $\mathbb{R}^3$: the Lotka-Volterra System, Physica A, 163 (1990), 683-722.  doi: 10.1016/0378-4371(90)90152-I.  Google Scholar [17] L. Haine, Geodesic flow on SO(4) and abelian surfaces, Math. Ann., 263 (1983), 435-472.  doi: 10.1007/BF01457053.  Google Scholar [18] D. D. Holm, Geometric Mechanics. Part I: Dynamics and Symmetry. Part II: Rotating, Translating and Rolling., Imperial College Press, 1$^{st}$ edition 2008, 2$^{nd}$ edition, 2011. doi: 10.1142/p802.  Google Scholar [19] D. D. Holm, T. Schmah and C. Stoica, Geometric Mechanics and Symmetry. From Finite to Infinite Dimensions, Cambridge University Press, 2009.  Google Scholar [20] E. Husson, Recherches des intégrales algébriques dans le mouvement d'un corps pesant autour d'un point fixe, Ann. Fac. Sci., 8 (1906), 73-152.   Google Scholar [21] A. A. Iliukhin, Spacial Problems of Nonlinear Theory of Elastic Rods, Naukova Dumka, Kiev, 1979. (in Russian).  Google Scholar [22] Yu. Ilyashenko and S. Yakovenko, Lectures on Analytical Differential Equations, Graduate Studies in Math., 86 AMS, Providence, RI, 2008.  Google Scholar [23] V. V. Kozlov, Symmetries, Topology and Resonances in Hamiltonian Mechanics, Springer, 1996. doi: 10.1007/978-3-642-78393-7.  Google Scholar [24] E. Leimanis, The General Problem of the Motion of Coupled Rigid Bodies about a Fixed Point, Springer, 1965. doi: 10.1007/978-3-642-88412-2.  Google Scholar [25] Y. Z. Liu and Y. Xue, Formulation of Kirchhoff rod based on quasi-coordinates, Technische Mechanik, Band, 24 (2004), 206-210. Google Scholar [26] A. J. Maciejewski and S. I. Popov, Invariants of homogeneous ordinary differential equations, Reports on Mathematical Physics, 41 (1998), 287-310.  doi: 10.1016/S0034-4877(98)80017-7.  Google Scholar [27] A. J. Maciejewski, S. I. Popov and J.-M. Strelcyn, The Euler equations on Lie algebra so(4): An elementary approach to integrability condition, Journal of Mathematical Physics, 42 (2001), 2701-2717.  doi: 10.1063/1.1370550.  Google Scholar [28] A. J. Maciejewski and M. Przybylska, Differential Galois approach to the non-integrability of the heavy top problem, Annales de la Faculté des Sciences de Toulouse. Mathématiques, Série 6, 14 (2005), 123-160.  Google Scholar [29] J. E. Marsden, Lectures on Mechanics, London Math. Soc. Lecture Notes Series, 174, Cambridge University Press, 1992. doi: 10.1017/CBO9780511624001.  Google Scholar [30] J. Montaldi and T. Ratiu (Eds), Geometric Mechanics and Symmetry: The Peyresq Lectures, London Math. Soc. Lecture Notes Series, 306, Cambridge University Press, 2005. doi: 10.1017/CBO9780511526367.  Google Scholar [31] R. Narasimhan, Analysis on Real and Complex Manifolds, 3$^{rd}$ printing, North-Holland, Amsterdam-New York-Oxford, 1985.  Google Scholar [32] P. J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Math., 107, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-1-4684-0274-2.  Google Scholar [33] A. M. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras, Ⅰ, Birkhauser Verlag, Basel, 1990. doi: 10.1007/978-3-0348-9257-5.  Google Scholar [34] P. Ya. Polubarinova-Kochina, On uniform solutions and algebraic integrals of the problem about rotation of the heavy rigid body problem around a fixed point, Motion of the Rigid Body around a fixed Point. S. V. Kovalevskaya Memorial Volume, Edition of Academy of Sciences of USSR, Moscow-Leningrad, (1940), 157-186. Google Scholar [35] S. I. Popov, On the existence of depending on p; q; r; γ first integral of the Euler-Poisson system, Theoretical and Applied Mechanics, Bulgarian Academy of Sciences, 2 (1981), 28-33. (in Bulgarian). Google Scholar [36] S. I. Popov, On the nonexistence of a new first integral $F(p,q,r,γ,γ') = const$ of the problem of a heavy rigid body motion about a fixed point, Comptes rendus de l'Académie bulgare des Sciences, 38 (1985), 583-586.   Google Scholar [37] S. I. Popov, On the nonexistence of a new first integral $F(p, q, r, γ, γ') = const$ of the problem of a heavy rigid body motion about a fixed point, Theoretical and Applied Mechanics, Bulgarian Academy of Sciences, 4 (1988), 17-23. (in Russian). Google Scholar [38] S. I. Popov and J.-M. Strelcyn, On rational integrability of Euler equations on Lie algebra so(4, C), Israel Journal of Mathemetics, 163 (2008), 263-283.  doi: 10.1007/s11856-008-0012-7.  Google Scholar [39] S. I. Popov and J.-M. Strelcyn, The Euler-Poisson equations: an elementary approach to partial integrability conditions, (in preparation). Google Scholar [40] B. V. Shabat, Introduction to Complex Analysis, Part II. Functions of Several Variables, Translations of Mathematical Monographs, 110, American Mathematical Society, Providence, RI, 1992.  Google Scholar [41] V. V. Trofimov, Introduction to the Geometry of Manifolds with Symmetry, Mathematics and its Applications, 270, Kluwer Academic Publishers Group, Dodrecht, 1994. doi: 10.1007/978-94-017-1961-2.  Google Scholar [42] S. L. Ziglin, Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. Ⅰ, Akademiya Nauk SSSR. Funktsional'nyĭ Analiz i ego Prilozheniya, 16 (1982), 30-41, (Russian) [Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. Ⅰ], Functional Anal. Appl., 16 (1982), 181-189. doi: 10.1007/BF01081586.  Google Scholar [43] S. L. Ziglin, Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. Ⅱ, Akademiya Nauk SSSR. Funktsional'nyĭ Analiz i ego Prilozheniya, 17 (1983), 8-23, (Russian) [Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. Ⅱ], Functional Anal. Appl., 17 (1983), 8-23.  Google Scholar [44] S. L. Ziglin, On the absence of a real-analytic first integral in some problems in dynamics, Funktsional. Anal. i Prilozhen., 31 (1997), 3-11, (Russian) [On the absence of a real-analytic first integral in some problems in dynamics], Functional Anal. Appl., 31 (1997), 3-9. doi: 10.1007/BF02465998.  Google Scholar

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##### References:
 [1] M. Adler and P. van Moerbeke, The algebraic integrability of geodesic flow on SO(4), Invent. Math., 67 (1982), 297-331.  doi: 10.1007/BF01393820.  Google Scholar [2] Yu. A. Arkhangelskii, Analytical Dynamics of the Rigid Body, Nauka, Moscow, 1977. (in Russian).  Google Scholar [3] Yu. A. Arkhangelskii, On one new property of Euler-Poisson equations, Doklady Akad. Nauk USSR, 258 (1981), 810-811. (in Russian).  Google Scholar [4] V. I. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Math., 60, 2$^{nd}$ edition, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar [5] M. Audin, Spinning Tops. A Course on Integrable Systems, Cambridge Studies in Advanced Mathematics, 51, Cambridge University Press, 1996.  Google Scholar [6] M. Audin, Les Systèmes Hamiltoniens et Leur Intégrabilité, Cours spécialisé, 8, Société Mathématique de France, EDP Sciences, 2001, (French) [Hamiltonian Systems and Their Integrability], SMF/AMS Texts and Monographs, 15, American Mathematical Society, 2008.  Google Scholar [7] O. I. Bogoyavlenskii, Euler equations on finite-dimensional Lie coalgebras, arising in problems of mathematical physics, Uspekhi Mat. Nauk, 47 (1992), 107-46, (Russian) [Euler equations on finite-dimensional Lie coalgebras, arising in problems of mathematical physics], Russian Math. Surveys, 47 (1992), 117-189. doi: 10.1070/RM1992v047n01ABEH000863.  Google Scholar [8] O. I. Bogoyavlenskii, Integrable Euler equations on Lie algebras arising in problems of mathematical physics, Izv. Akad. Nauk SSSR Ser. Mat., 48 (1984), 883-938, (Russian) [Integrable Euler equations on Lie algebras arising in problems of mathematical physics], Math. USSR Izvestiya, 25 (1985), 207-257. doi: 10.1070/IM1985v025n02ABEH001278.  Google Scholar [9] O. I. Bogoyavlenskii, Integrable Euler equations on six-dimensional Lie algebras, Doklady Akad. Nauk USSR, 268 (1983), 11-15, (Russian) [Integrable Euler equations on sixdimensional Lie algebras], Soviet Math. Doklady, 27 (1983), 1-5.  Google Scholar [10] A. V. Borisov and I. S. Mamaev, Poisson Structures and Lie Algebras in Hamiltonian Mechanics, Library "R & C Dynamics", Edited by Izdatel'skiĭ Dom "Udmurtskiĭ Universitet", Izhevsk, 1999. (in Russian). http://ics.org.ru/publications/index.php?cat=103&author=23  Google Scholar [11] A. V. Borisov and I. S. Mamaev, Rigid Body Dynamics. Hamiltonian Methods, Integrability, Chaos, Moscow-Izhevsk: Institute of Computer Science, 2005. http://ics.org.ru/publications/index.php?cat=103&author=23 Google Scholar [12] A. V. Borisov and I. S. Mamaev, Modern Methods of the Theory of Integrable Systems, Moscow-Izhevsk: Institute of Computer Science, 2003. http://ics.org.ru/publications/index.php?cat=103&author=23  Google Scholar [13] A. T. Fomenko, Integrability and Nonintegrability in Geometry and Mechanics, Dodrecht: Kluwer Academic Publishers, 1988. doi: 10.1007/978-94-009-3069-8.  Google Scholar [14] I. N. Ganshenko, G. V. Gorr and A. M. Kovalev, Classical Problems of Rigid Body Dynamics, Kiev, Naukova Dumka, 2012. (in Russian). Google Scholar [15] V. V. Golubev, Lectures on Integration of the Equations of Motion of a Rigid Body about a Fixed Point, Moscow, Gostekhizdat, 1953, (Russian) [Lectures on Integration of the Equations of Motion of a Rigid Body about a Fixed Point], Israel program for scientific translations, Haifa, 1960.  Google Scholar [16] B. Grammaticos, J. Moulin-Ollagnier, A. Ramani, J.-M. Strelcyn and S. Wojciechowski, Integrals of quadratic ordinary differential equations in $\mathbb{R}^3$: the Lotka-Volterra System, Physica A, 163 (1990), 683-722.  doi: 10.1016/0378-4371(90)90152-I.  Google Scholar [17] L. Haine, Geodesic flow on SO(4) and abelian surfaces, Math. Ann., 263 (1983), 435-472.  doi: 10.1007/BF01457053.  Google Scholar [18] D. D. Holm, Geometric Mechanics. Part I: Dynamics and Symmetry. Part II: Rotating, Translating and Rolling., Imperial College Press, 1$^{st}$ edition 2008, 2$^{nd}$ edition, 2011. doi: 10.1142/p802.  Google Scholar [19] D. D. Holm, T. Schmah and C. Stoica, Geometric Mechanics and Symmetry. From Finite to Infinite Dimensions, Cambridge University Press, 2009.  Google Scholar [20] E. Husson, Recherches des intégrales algébriques dans le mouvement d'un corps pesant autour d'un point fixe, Ann. Fac. Sci., 8 (1906), 73-152.   Google Scholar [21] A. A. Iliukhin, Spacial Problems of Nonlinear Theory of Elastic Rods, Naukova Dumka, Kiev, 1979. (in Russian).  Google Scholar [22] Yu. Ilyashenko and S. Yakovenko, Lectures on Analytical Differential Equations, Graduate Studies in Math., 86 AMS, Providence, RI, 2008.  Google Scholar [23] V. V. Kozlov, Symmetries, Topology and Resonances in Hamiltonian Mechanics, Springer, 1996. doi: 10.1007/978-3-642-78393-7.  Google Scholar [24] E. Leimanis, The General Problem of the Motion of Coupled Rigid Bodies about a Fixed Point, Springer, 1965. doi: 10.1007/978-3-642-88412-2.  Google Scholar [25] Y. Z. Liu and Y. Xue, Formulation of Kirchhoff rod based on quasi-coordinates, Technische Mechanik, Band, 24 (2004), 206-210. Google Scholar [26] A. J. Maciejewski and S. I. Popov, Invariants of homogeneous ordinary differential equations, Reports on Mathematical Physics, 41 (1998), 287-310.  doi: 10.1016/S0034-4877(98)80017-7.  Google Scholar [27] A. J. Maciejewski, S. I. Popov and J.-M. Strelcyn, The Euler equations on Lie algebra so(4): An elementary approach to integrability condition, Journal of Mathematical Physics, 42 (2001), 2701-2717.  doi: 10.1063/1.1370550.  Google Scholar [28] A. J. Maciejewski and M. Przybylska, Differential Galois approach to the non-integrability of the heavy top problem, Annales de la Faculté des Sciences de Toulouse. Mathématiques, Série 6, 14 (2005), 123-160.  Google Scholar [29] J. E. Marsden, Lectures on Mechanics, London Math. Soc. Lecture Notes Series, 174, Cambridge University Press, 1992. doi: 10.1017/CBO9780511624001.  Google Scholar [30] J. Montaldi and T. Ratiu (Eds), Geometric Mechanics and Symmetry: The Peyresq Lectures, London Math. Soc. Lecture Notes Series, 306, Cambridge University Press, 2005. doi: 10.1017/CBO9780511526367.  Google Scholar [31] R. Narasimhan, Analysis on Real and Complex Manifolds, 3$^{rd}$ printing, North-Holland, Amsterdam-New York-Oxford, 1985.  Google Scholar [32] P. J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Math., 107, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-1-4684-0274-2.  Google Scholar [33] A. M. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras, Ⅰ, Birkhauser Verlag, Basel, 1990. doi: 10.1007/978-3-0348-9257-5.  Google Scholar [34] P. Ya. Polubarinova-Kochina, On uniform solutions and algebraic integrals of the problem about rotation of the heavy rigid body problem around a fixed point, Motion of the Rigid Body around a fixed Point. S. V. Kovalevskaya Memorial Volume, Edition of Academy of Sciences of USSR, Moscow-Leningrad, (1940), 157-186. Google Scholar [35] S. I. Popov, On the existence of depending on p; q; r; γ first integral of the Euler-Poisson system, Theoretical and Applied Mechanics, Bulgarian Academy of Sciences, 2 (1981), 28-33. (in Bulgarian). Google Scholar [36] S. I. Popov, On the nonexistence of a new first integral $F(p,q,r,γ,γ') = const$ of the problem of a heavy rigid body motion about a fixed point, Comptes rendus de l'Académie bulgare des Sciences, 38 (1985), 583-586.   Google Scholar [37] S. I. Popov, On the nonexistence of a new first integral $F(p, q, r, γ, γ') = const$ of the problem of a heavy rigid body motion about a fixed point, Theoretical and Applied Mechanics, Bulgarian Academy of Sciences, 4 (1988), 17-23. (in Russian). Google Scholar [38] S. I. Popov and J.-M. Strelcyn, On rational integrability of Euler equations on Lie algebra so(4, C), Israel Journal of Mathemetics, 163 (2008), 263-283.  doi: 10.1007/s11856-008-0012-7.  Google Scholar [39] S. I. Popov and J.-M. Strelcyn, The Euler-Poisson equations: an elementary approach to partial integrability conditions, (in preparation). Google Scholar [40] B. V. Shabat, Introduction to Complex Analysis, Part II. Functions of Several Variables, Translations of Mathematical Monographs, 110, American Mathematical Society, Providence, RI, 1992.  Google Scholar [41] V. V. Trofimov, Introduction to the Geometry of Manifolds with Symmetry, Mathematics and its Applications, 270, Kluwer Academic Publishers Group, Dodrecht, 1994. doi: 10.1007/978-94-017-1961-2.  Google Scholar [42] S. L. Ziglin, Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. Ⅰ, Akademiya Nauk SSSR. Funktsional'nyĭ Analiz i ego Prilozheniya, 16 (1982), 30-41, (Russian) [Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. Ⅰ], Functional Anal. Appl., 16 (1982), 181-189. doi: 10.1007/BF01081586.  Google Scholar [43] S. L. Ziglin, Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. Ⅱ, Akademiya Nauk SSSR. Funktsional'nyĭ Analiz i ego Prilozheniya, 17 (1983), 8-23, (Russian) [Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. Ⅱ], Functional Anal. Appl., 17 (1983), 8-23.  Google Scholar [44] S. L. Ziglin, On the absence of a real-analytic first integral in some problems in dynamics, Funktsional. Anal. i Prilozhen., 31 (1997), 3-11, (Russian) [On the absence of a real-analytic first integral in some problems in dynamics], Functional Anal. Appl., 31 (1997), 3-9. doi: 10.1007/BF02465998.  Google Scholar
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