On the Poincar\'{e} problem of the third integral of the equations of rotation of a heavy asymmetric top

V. V. Kozlov

Izvestiya. Mathematics , Tome 88 (2024) no. 6, pp. 1173-1184.

Résumé

The Poincaré third-integral problem for the equations of rotation of a heavy asymmetric rigid body with fixed point is considered; the third integral is assumed to be independent of the energy and area integrals, and can be represented as a series in powers of a small parameter whose coefficients are single-valued analytic functions on the six-dimensional phase space. The small parameter here is the ratio of the distance from the centre of mass to the point of support to the characteristic size of the rigid body. This problem, which was posed by Poincaré in the fifth chapter of his celebrated New methods of celestial mechanics, has a negative solution under the additional assumption that this third integral is in involution with the area integral (this result was obtained by the author of the present note already in 1975). In the present paper, the Poincaré problem is solved in its original general problem statement (without the assumption that the Poisson bracket should vanish) – it is shown that if the body is dynamically asymmetric, then the third single-valued analytic integral does not exist. The proof depends on the Poincaré method augmented with some novel ideas and a more careful analysis of the expansion of the perturbing function as a Fourier series in the angle variables.

Détail

Keywords: asymmetric top, Hamiltonian system, symmetry group, secular set, key set, analytic integral, polynomial integral, reduced system.
Mots-clés : action–angle variables

Bibliographie

[1] H. Poincaré, Les méthodes nouvelles de la mécanique céleste, v. I, Gauthier-Villars, Paris, 1892 | MR | Zbl

[2] Ed. Husson, “Sur un théorème de M. Poincaré, relativement au mouvement d'un solide pesant”, Acta Math., 31:1 (1908), 71–88 | DOI | MR | Zbl

[3] V. V. Kozlov, “The geometry of the “action–angle” variables in the Euler–Poinsot problem”, Vestnik Moskov. Univ. Ser. I Mat. Meh., 29:5 (1974), 74–79 (Russian) | MR | Zbl

[4] V. V. Kozlov, “Non-existence of an additional analytic integral in the problem of the motion of a unsymmetric heavy solid about a fixed point”, Vestnik Moskov. Univ. Ser. I Mat. Meh., 30:1 (1975), 105–110 (Russian) | MR | Zbl

[5] V. V. Kozlov, Methods of qualitative analysis in rigid body dynamics, 2nd ed., Regulyarnaya i Khaotichaskaya Dinamika, Moscow–Izhevsk, 2000 (Russian) | MR | Zbl

[6] S. L. Ziglin, “Branching of solutions and nonexistence of first integrals in Hamiltonian mechanics. I”, Funct. Anal. Appl., 16:3 (1982), 181–189 ; II,, 17:1 (1983), 6–17 | DOI | DOI

[7] V. V. Kozlov, “Nonexistence of analytic integrals near equilibrium positions of Hamiltonian systems”, Moscow Univ. Mech. Bull., 31 (1976), 35–39

[8] T. V. Sal'nikova, “Nonintegrability of perturbed Lagrange problem”, Moscow Univ. Mech. Bull., 39:4 (1984), 32–39

[9] V. V. Kozlov and D. V. Treshchev, “Nonintegrability of general problem of rotation of a dynamically symmetrical heavy rigid body with a fixed point”, Moscow Univ. Mech. Bull., 41:1 (1986), 11–16 | Zbl

[10] S. A. Dovbysh, “Splitting of separatrices for unstable uniform rotations and nonintegrability of perturbed Lagrange problem”, Moscow Univ. Mech. Bull., 45:3 (1990), 33–39 | Zbl

[11] S. V. Bolotin, “Variational methods for constructing chaotic motions in rigid-body dynamics”, J. Appl. Math. Mech., 56:2 (1992), 198–205 | DOI | Zbl

[12] S. L. Ziglin, “The absence of an additional real-analytic first integral in some problems of dynamics”, Funct. Anal. Appl., 31:1 (1997), 3–9 | DOI

[13] A. V. Borisov and I. S. Mamaev, Rigid body dynamics Hamiltonian methods, integrability, chaos, 2nd ed., Regulyarnaya i Khaotichaskaya Dinamika, Moscow–Izhevsk, 2005 (Russian) | MR | Zbl

[14] Éd. Husson, “Recherche des intégrales algébriques dans le mouvement d'un solide pesant autour d'un point fixe”, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (2), 8 (1906), 73–152 | DOI | MR | Zbl

[15] A. I. Dokshevich, Closed-form solutions of the Euler–Poisson equations, Naukova Dumka, Kiev, 1992 (Russian) | MR