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.

[2]

Yu. A. Arkhangelskii, Analytical Dynamics of the Rigid Body, Nauka, Moscow, 1977. (in Russian).

[3]

Yu. A. Arkhangelskii, On one new property of Euler-Poisson equations, Doklady Akad. Nauk USSR, 258 (1981), 810-811. (in Russian).

[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.

[5]

M. Audin, Spinning Tops. A Course on Integrable Systems, Cambridge Studies in Advanced Mathematics, 51, Cambridge University Press, 1996.

[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.

[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.

[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.

[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.

[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

[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

[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

[13]

A. T. Fomenko, Integrability and Nonintegrability in Geometry and Mechanics, Dodrecht: Kluwer Academic Publishers, 1988. doi: 10.1007/978-94-009-3069-8.

[14]

I. N. Ganshenko, G. V. Gorr and A. M. Kovalev, Classical Problems of Rigid Body Dynamics, Kiev, Naukova Dumka, 2012. (in Russian).

[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.

[16]

B. GrammaticosJ. Moulin-OllagnierA. RamaniJ.-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.

[17]

L. Haine, Geodesic flow on SO(4) and abelian surfaces, Math. Ann., 263 (1983), 435-472. doi: 10.1007/BF01457053.

[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.

[19]

D. D. Holm, T. Schmah and C. Stoica, Geometric Mechanics and Symmetry. From Finite to Infinite Dimensions, Cambridge University Press, 2009.

[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.

[21]

A. A. Iliukhin, Spacial Problems of Nonlinear Theory of Elastic Rods, Naukova Dumka, Kiev, 1979. (in Russian).

[22]

Yu. Ilyashenko and S. Yakovenko, Lectures on Analytical Differential Equations, Graduate Studies in Math., 86 AMS, Providence, RI, 2008.

[23]

V. V. Kozlov, Symmetries, Topology and Resonances in Hamiltonian Mechanics, Springer, 1996. doi: 10.1007/978-3-642-78393-7.

[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.

[25]

Y. Z. Liu and Y. Xue, Formulation of Kirchhoff rod based on quasi-coordinates, Technische Mechanik, Band, 24 (2004), 206-210.

[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.

[27]

A. J. MaciejewskiS. 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.

[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.

[29]

J. E. Marsden, Lectures on Mechanics, London Math. Soc. Lecture Notes Series, 174, Cambridge University Press, 1992. doi: 10.1017/CBO9780511624001.

[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.

[31]

R. Narasimhan, Analysis on Real and Complex Manifolds, 3$^{rd}$ printing, North-Holland, Amsterdam-New York-Oxford, 1985.

[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.

[33]

A. M. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras, , Birkhauser Verlag, Basel, 1990. doi: 10.1007/978-3-0348-9257-5.

[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.

[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).

[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.

[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).

[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.

[39]

S. I. Popov and J.-M. Strelcyn, The Euler-Poisson equations: an elementary approach to partial integrability conditions, (in preparation).

[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.

[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.

[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.

[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.

[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.

show all references

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.

[2]

Yu. A. Arkhangelskii, Analytical Dynamics of the Rigid Body, Nauka, Moscow, 1977. (in Russian).

[3]

Yu. A. Arkhangelskii, On one new property of Euler-Poisson equations, Doklady Akad. Nauk USSR, 258 (1981), 810-811. (in Russian).

[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.

[5]

M. Audin, Spinning Tops. A Course on Integrable Systems, Cambridge Studies in Advanced Mathematics, 51, Cambridge University Press, 1996.

[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.

[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.

[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.

[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.

[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

[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

[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

[13]

A. T. Fomenko, Integrability and Nonintegrability in Geometry and Mechanics, Dodrecht: Kluwer Academic Publishers, 1988. doi: 10.1007/978-94-009-3069-8.

[14]

I. N. Ganshenko, G. V. Gorr and A. M. Kovalev, Classical Problems of Rigid Body Dynamics, Kiev, Naukova Dumka, 2012. (in Russian).

[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.

[16]

B. GrammaticosJ. Moulin-OllagnierA. RamaniJ.-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.

[17]

L. Haine, Geodesic flow on SO(4) and abelian surfaces, Math. Ann., 263 (1983), 435-472. doi: 10.1007/BF01457053.

[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.

[19]

D. D. Holm, T. Schmah and C. Stoica, Geometric Mechanics and Symmetry. From Finite to Infinite Dimensions, Cambridge University Press, 2009.

[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.

[21]

A. A. Iliukhin, Spacial Problems of Nonlinear Theory of Elastic Rods, Naukova Dumka, Kiev, 1979. (in Russian).

[22]

Yu. Ilyashenko and S. Yakovenko, Lectures on Analytical Differential Equations, Graduate Studies in Math., 86 AMS, Providence, RI, 2008.

[23]

V. V. Kozlov, Symmetries, Topology and Resonances in Hamiltonian Mechanics, Springer, 1996. doi: 10.1007/978-3-642-78393-7.

[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.

[25]

Y. Z. Liu and Y. Xue, Formulation of Kirchhoff rod based on quasi-coordinates, Technische Mechanik, Band, 24 (2004), 206-210.

[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.

[27]

A. J. MaciejewskiS. 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.

[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.

[29]

J. E. Marsden, Lectures on Mechanics, London Math. Soc. Lecture Notes Series, 174, Cambridge University Press, 1992. doi: 10.1017/CBO9780511624001.

[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.

[31]

R. Narasimhan, Analysis on Real and Complex Manifolds, 3$^{rd}$ printing, North-Holland, Amsterdam-New York-Oxford, 1985.

[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.

[33]

A. M. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras, , Birkhauser Verlag, Basel, 1990. doi: 10.1007/978-3-0348-9257-5.

[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.

[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).

[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.

[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).

[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.

[39]

S. I. Popov and J.-M. Strelcyn, The Euler-Poisson equations: an elementary approach to partial integrability conditions, (in preparation).

[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.

[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.

[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.

[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.

[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.

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