Integrable cases of the dynamics of a~rigid body, and integrable systems on the spheres~$S^n$

O. I. Bogoyavlenskii

Izvestiya. Mathematics , Tome 27 (1986) no. 2, pp. 203-218.

Résumé

Infinite-dimensional sets of integrable cases of the equations of rotation of a rigid body about a fixed point in a central force field, and also in more complicated fields (in the presence of symmetries of the inertia tensor), are found. Also considered are infinite-dimensional sets of integrable cases of the dynamics of a particle in potential fields in Euclidean space on $n$-axial ellipsoids and the spheres $S^n$. Bibliography: 34 titles.

Bibliographie

[1] Golubev V. V., Lektsii po integrirovaniyu uravnenii dvizheniya tyazhelogo tverdogo tela okolo nepodvizhnoi tochki, Gostekhizdat, M., 1953

[2] Jacobi C. G. J., Vorlesungen über Dynamik, Königsberg, 1866 | Zbl

[3] Moser J., “Geometry of Quadrics and Spectral Theory”, The Chern Symposium 1979 (Proc. Internat. Sympos., Berkeley, Calif., 1979), Springer-Verlag, 1979 | MR | Zbl

[4] Moser J., Integrable Hamiltonian Systems and Spectral Theory, Lezioni Fermiane, Pisa, 1983 | MR

[5] Moser J., “Various aspects of integrable Hamiltonian systems”, Dynamical systems (C.I.M.E. Summer School, Bressanone, 1978), Progr. Math., 8, Birkhäuser Boston, Mass., 1980, 233–289 | MR

[6] Knörrer H., “Geodesies on the Ellipsoid”, Invent. Math., 59:2 (1980), 119–145 | DOI | MR

[7] Neumann C., “De problemate quodam mechanico, quod ad priman integralium ultraellipticorum classen revocatur”, J. Reine Angew. Math., 56 (1859), 46–63 | Zbl

[8] Kosochatius E., Über Bewegungen eines Punktes, Dissertation at Univ. Göttingen, Druck von Gebr. Unger, Berlin, 1877

[9] Stäckel P., “Über die Bewegung eines Punktes in einer $n$-fachen Mannigfaltigkeit”, Math. Ann., 42 (1893), 537–563 | DOI | MR

[10] Bogoyavlenskii O. I., “Dva integriruemykh sluchaya dinamiki tverdogo tela v silovom pole”, Dokl. AN SSSR, 275:6 (1984), 1359–1363 | MR | Zbl

[11] Bogoyavlenskii O. I., “Integriruemye uravneniya Eilera na algebrakh Li, voznikayuschie v zadachakh matematicheskoi fiziki”, Izv. AN SSSR. Seriya matem., 48:5 (1984), 883–938 | MR

[12] Clebsch A., “Veber die Bewegung eines Körpers in einer Flüssigkeit”, Math. Ann., 3 (1871), 238–262 | DOI | MR

[13] Dirac P. A. M., “The Theory of Magnetic Poles”, Physics Rev., 74:7 (1948), 817–830 | DOI | MR | Zbl

[14] Novikov S. P., Shmeltser I., “Periodicheskie resheniya uravneniya Kirkhgofa dlya svobodnogo dvizheniya tverdogo tela v zhidkosti i rasshirennaya teoriya Lyusternika–Shnirelmana–Morsa (LMSh). I”, Funkts. analiz i ego prilozh., 15:3 (1981), 54–66 | MR

[15] Novikov S. P., “Gamiltonov formalizm i mnogoznachnyi analog teorii Morsa”, Uspekhi matem. nauk, 37:5 (1982), 3–49 | MR | Zbl

[16] Veselov A. P., “Uravnenie Landau–Lifshitsa i integriruemye sistemy klassicheskoi mekhaniki”, Dokl. AN SSSR, 270:5 (1983), 1094–1097 | MR

[17] Kötter F., “Veber die Bewegung eines festen Körpers in einer Flüssigkeit. I, II”, J. Reine Angew. Math., 109 (1892), 51–81, 89–111

[18] Steklov V. A., O dvizhenii tverdogo tela v zhidkosti, tip. Darre, Kharkov, 1893

[19] Morera G., “Sulla Separazione delle variabili”, Atti della R. Accademia delle Scienze di Torino, 16 (1881), 276–295

[20] Eleonskii V. M., Kulagin N. E., “O novykh sluchayakh integriruemosti uravnenii Landau–Lifshitsa”, ZhETF, 84:2 (1983), 616–629 | MR

[21] Levi-Civita T., Amaldi V., Lezioni di Meccanica Razionale, v. 2, Bologna, 1927

[22] Arnold V. I., Matematicheskie metody klassicheskoi mekhaniki, Nauka, M., 1974 | MR

[23] Kozlov V. V., Metody kachestvennogo analiza v dinamike tverdogo tela, MGU, M., 1980 | MR

[24] Brun F., “Rotation kring fix punkt”, Öfversigt at Kongl. Svenska Vetenskaps Akad. Förhadl, 7, Stokholm, 1893, 455–468

[25] Zhelobenko D. P., Kompaktnye gruppy Li i ikh predstavleniya, Nauka, M., 1970 | MR | Zbl

[26] Bogoyavlenskii O. Ya., “Novaya integriruemaya zadacha klassicheskoi mekhaniki”, Trudy III Mezhdunarodnogo simpoziuma po izbrannym problemam statisticheskoi mekhaniki (Dubna, avgust 1984), 165–170

[27] Lunev V. V., “Gidrodinamicheskaya analogiya zadachi o dvizhenii tverdogo tela s nepodvizhnoi tochkoi v pole sil Lorentsa”, Dokl. AN SSSR, 276:2 (1984), 351–355 | MR | Zbl

[28] Reiman A. G., “Integriruemye gamiltonovy sistemy, svyazannye s graduirovannymi algebrami Li”, Zapiski nauchnykh seminarov LOMI, 95, 1980, 3–55 | MR

[29] Wojciechowski S., “Integrability of one particle in a perturbated central quartic potential”, Physica Scripta, 31:6 (1985), 433–438 | DOI | MR | Zbl

[30] Goryachev D. N., “Novye sluchai dvizheniya tverdogo tela vokrug nepodvizhnoi tochki”, Izvestiya Varshavsk. universiteta, 1915, no. 3, 3–11

[31] Goryachev D. N., “Novye sluchai integriruemosti dinamicheskikh uravnenii Eilera”, Izvestiya Varshavsk. universiteta, 1916, no. 3, 3–15

[32] Brailov A. V., “Polnaya integriruemost nekotorykh geodezicheskikh potokov”, Dokl. AN SSSR, 271:2 (1983), 273–276 | MR

[33] Wojciechowski S., Two families of integrable one-particle potentials which remain integrable when the motion is restricted to a sphere or an ellipsoid, preprint UMIST, october 1984

[34] Wojciechowski S., Three families of integrable one particle potentials, preprint No 440 Universita di Roma, february 1985