Invariants of homogeneous dynamic systems of arbitrary odd order with dissipation. II. Fifth-order systems

M. V. Shamolin

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXV", Voronezh, April 26-30, 2024, Part 3, Tome 237 (2024), pp. 49-75.

Résumé

In this paper, we present new examples of integrable dynamical systems of the fifth order that are homogeneous in part of the variables. In these systems, subsystems on the tangent bundles of lower-dimensional manifolds can be distinguished. In the cases considered, the force field is partitioned into an internal (conservative) part and an external part. The external force introduced by a certain unimodular transformation has alternate dissipation; it is a generalization of fields examined earlier. Complete sets of first integrals and invariant differential forms are presented. The first part of the paper: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 236 (2024), pp. 72–88.

Détail

Keywords: dynamical system, integrability, dissipation, first integral with essential singular points, invariant differential form

Bibliographie

[1] Burbaki N., Integrirovanie. Mery, integrirovanie mer, Nauka, M., 1967

[2] Burbaki N., Integrirovanie. Mery na lokalno kompaktnykh prostranstvakh. Prodolzhenie mery. Integrirovanie mer. Mery na otdelimykh prostranstvakh, Nauka, M., 1977

[3] Veil G., Simmetriya, URSS, M., 2007

[4] Georgievskii D. V., Shamolin M. V., “Kinematika i geometriya mass tverdogo tela s nepodvizhnoi tochkoi v $\mathbb{R}^n$”, Dokl. RAN., 380:1 (2001), 47–50

[5] Georgievskii D. V., Shamolin M. V., “Obobschennye dinamicheskie uravneniya Eilera dlya tverdogo tela s nepodvizhnoi tochkoi v $\mathbb{R}^n$”, Dokl. RAN., 383:5 (2002), 635–637

[6] Georgievskii D. V., Shamolin M. V., “Pervye integraly uravnenii dvizheniya obobschennogo giroskopa v $\mathbb{R}^{n}$”, Vestn. Mosk. un-ta. Ser. 1. Mat. Mekh., 5 (2003), 37–41

[7] Dubrovin B. A., Novikov S. P., Fomenko A. T., Sovremennaya geometriya, Nauka, M., 1979

[8] Eroshin V. A., Samsonov V. A., Shamolin M. V., “Modelnaya zadacha o tormozhenii tela v soprotivlyayuscheisya srede pri struinom obtekanii”, Izvyu RAN. Mekh. zhidk. gaza., 1995, no. 3, 23–27

[9] Ivanova T. A., “Ob uravneniyakh Eilera v modelyakh teoreticheskoi fiziki”, Mat. zametki., 52:2 (1992), 43–51

[10] Kamke E., Spravochnik po obyknovennym differentsialnym uravneniyam, Nauka, M., 1971

[11] Klein F., Neevklidova geometriya, URSS, M., 2017

[12] Kozlov V. V., “Integriruemost i neintegriruemost v gamiltonovoi mekhanike”, Usp. mat. nauk., 38:1 (1983), 3–67

[13] Kozlov V. V., “Ratsionalnye integraly kvaziodnorodnykh dinamicheskikh sistem”, Prikl. mat. mekh., 79:3 (2015), 307–316

[14] Kozlov V. V., “Tenzornye invarianty i integrirovanie differentsialnykh uravnenii”, Usp. mat. nauk., 74:1 (445) (2019), 117–148

[15] Kolmogorov A. N., “O dinamicheskikh sistemakh s integralnym invariantom na tore”, Dokl. AN SSSR., 93:5 (1953), 763–766

[16] Pokhodnya N. V., Shamolin M. V., “Novyi sluchai integriruemosti v dinamike mnogomernogo tela”, Vestn. SamGU. Estestvennonauch. ser., 9:100 (2012), 136–150

[17] Pokhodnya N. V., Shamolin M. V., “Nekotorye usloviya integriruemosti dinamicheskikh sistem v transtsendentnykh funktsiyakh”, Vestn. SamGU. Estestvennonauch. ser., 9/1:110 (2013), 35–41

[18] Pokhodnya N. V., Shamolin M. V., “Integriruemye sistemy na kasatelnom rassloenii k mnogomernoi sfere”, Vestn. SamGU. Estestvennonauch. ser., 7:118 (2014), 60–69

[19] Samsonov V. A., Shamolin M. V., “K zadache o dvizhenii tela v soprotivlyayuscheisya srede”, Vestn. Mosk. un-ta. Ser. 1. Mat. mekh., 1989, no. 3, 51–54

[20] Trofimov V. V., “Uravneniya Eilera na konechnomernykh razreshimykh gruppakh Li”, Izv. AN SSSR. Ser. mat., 44:5 (1980), 1191–1199

[21] Trofimov V. V., “Simplekticheskie struktury na gruppakh avtomorfizmov simmetricheskikh prostranstv”, Vestn. Mosk. un-ta. Ser. 1. Mat. Mekh., 1984, no. 6, 31–33

[22] Trofimov V. V., Fomenko A. T., “Metodika postroeniya gamiltonovykh potokov na simmetricheskikh prostranstvakh i integriruemost nekotorykh gidrodinamicheskikh sistem”, Dokl. AN SSSR., 254:6 (1980), 1349–1353

[23] Trofimov V. V., Shamolin M. V., “Geometricheskie i dinamicheskie invarianty integriruemykh gamiltonovykh i dissipativnykh sistem”, Fundam. prikl. mat., 16:4 (2010), 3–229

[24] Shabat B. V., Vvedenie v kompleksnyi analiz, Nauka, M., 1987

[25] Shamolin M. V., “Ob integriruemosti v transtsendentnykh funktsiyakh”, Usp. mat. nauk., 53:3 (1998), 209–210

[26] Shamolin M. V., “Novye integriruemye po Yakobi sluchai v dinamike tverdogo tela, vzaimodeistvuyuschego so sredoi”, Dokl. RAN., 364:5 (1999), 627–629

[27] Shamolin M. V., “Integriruemost po Yakobi v zadache o dvizhenii chetyrekhmernogo tverdogo tela v soprotivlyayuscheisya srede”, Dokl. RAN., 375:3 (2000), 343–346

[28] Shamolin M. V., “Ob integrirovanii nekotorykh klassov nekonservativnykh sistem”, Usp. mat. nauk., 57:1 (2002), 169–170

[29] Shamolin M. V., “Ob odnom integriruemom sluchae uravnenii dinamiki na $so(4)\times\mathbb{R}^{4}$”, Usp. mat. nauk., 60:6 (2005), 233–234

[30] Shamolin M. V., “Sopostavlenie integriruemykh po Yakobi sluchaev ploskogo i prostranstvennogo dvizheniya tela v srede pri struinom obtekanii”, Prikl. mat. mekh., 69:6 (2005), 1003–1010

[31] Shamolin M. V., “Sluchai polnoi integriruemosti v dinamike na kasatelnom rassloenii dvumernoi sfery”, Usp. mat. nauk., 62:5 (2007), 169–170

[32] Shamolin M. V., “Novye sluchai polnoi integriruemosti v dinamike dinamicheski simmetrichnogo chetyrekhmernogo tverdogo tela v nekonservativnom pole”, Dokl. RAN., 425:3 (2009), 338–342

[33] Shamolin M. V., “Sluchai polnoi integriruemosti v dinamike chetyrekhmernogo tverdogo tela v nekonservativnom pole”, Usp. mat. nauk., 65:1 (2010), 189–190

[34] Shamolin M. V., “Novyi sluchai integriruemosti v dinamike chetyrekhmernogo tverdogo tela v nekonservativnom pole”, Dokl. RAN., 437:2 (2011), 190–193

[35] Shamolin M. V., “Polnyi spisok pervykh integralov v zadache o dvizhenii chetyrekhmernogo tverdogo tela v nekonservativnom pole pri nalichii lineinogo dempfirovaniya”, Dokl. RAN., 440:2 (2011), 187–190

[36] Shamolin M. V., “Novyi sluchai integriruemosti v dinamike chetyrekhmernogo tverdogo tela v nekonservativnom pole pri nalichii lineinogo dempfirovaniya”, Dokl. RAN., 444:5 (2012), 506–509

[37] Shamolin M. V., “Novyi sluchai integriruemosti v prostranstvennoi dinamike tverdogo tela, vzaimodeistvuyuschego so sredoi, pri uchete lineinogo dempfirovaniya”, Dokl. RAN., 442:4 (2012), 479–481

[38] Shamolin M. V., “Novyi sluchai integriruemosti v dinamike mnogomernogo tverdogo tela v nekonservativnom pole”, Dokl. RAN., 453:1 (2013), 46–49

[39] Shamolin M. V., “Novyi sluchai integriruemosti uravnenii dinamiki na kasatelnom rassloenii k trekhmernoi sfere”, Usp. mat. nauk., 68:5 (413) (2013), 185–186

[40] Shamolin M. V., “Polnyi spisok pervykh integralov dinamicheskikh uravnenii dvizheniya chetyrekhmernogo tverdogo tela v nekonservativnom pole pri nalichii lineinogo dempfirovaniya”, Dokl. RAN., 449:4 (2013), 416–419

[41] Shamolin M. V., “Novyi sluchai integriruemosti v dinamike mnogomernogo tverdogo tela v nekonservativnom pole pri uchete lineinogo dempfirovaniya”, Dokl. RAN., 457:5 (2014), 542–545

[42] Shamolin M. V., “Integriruemye sistemy s peremennoi dissipatsiei na kasatelnom rassloenii k mnogomernoi sfere i prilozheniya”, Fundam. prikl. mat., 20:4 (2015), 3–231

[43] Shamolin M. V., “Polnyi spisok pervykh integralov dinamicheskikh uravnenii dvizheniya mnogomernogo tverdogo tela v nekonservativnom pole”, Dokl. RAN., 461:5 (2015), 533–536

[44] Shamolin M. V., “Polnyi spisok pervykh integralov uravnenii dvizheniya mnogomernogo tverdogo tela v nekonservativnom pole pri nalichii lineinogo dempfirovaniya”, Dokl. RAN., 464:6 (2015), 688–692

[45] Shamolin M. V., “Integriruemye nekonservativnye dinamicheskie sistemy na kasatelnom rassloenii k mnogomernoi sfere”, Differ. uravn., 52:6 (2016), 743–759

[46] Shamolin M. V., “Novye sluchai integriruemykh sistem s dissipatsiei na kasatelnom rassloenii dvumernogo mnogoobraziya”, Dokl. RAN., 475:5 (2017), 519–523

[47] Shamolin M. V., “Novye sluchai integriruemykh sistem s dissipatsiei na kasatelnom rassloenii k mnogomernoi sfere”, Dokl. RAN., 474:2 (2017), 177–181

[48] Shamolin M. V., “Novye sluchai integriruemykh sistem s dissipatsiei na kasatelnom rassloenii trekhmernogo mnogoobraziya”, Dokl. RAN., 477:2 (2017), 168–172

[49] Shamolin M. V., “Novye sluchai integriruemykh sistem s dissipatsiei na kasatelnom rassloenii mnogomernogo mnogoobraziya”, Dokl. RAN., 482:5 (2018), 527–533

[50] Shamolin M. V., “Novye sluchai integriruemykh sistem s dissipatsiei na kasatelnom rassloenii chetyrekhmernogo mnogoobraziya”, Dokl. RAN., 479:3 (2018), 270–276

[51] Shamolin M. V., “Novye sluchai integriruemykh sistem devyatogo poryadka s dissipatsiei”, Dokl. RAN., 489:6 (2019), 592–598

[52] Shamolin M. V., “Novye sluchai integriruemykh sistem pyatogo poryadka s dissipatsiei”, Dokl. RAN., 485:5 (2019), 583–587

[53] Shamolin M. V., “Novye sluchai integriruemykh sistem sedmogo poryadka s dissipatsiei”, Dokl. RAN., 487:4 (2019), 381–386

[54] Shamolin M. V., “Novye sluchai integriruemykh sistem nechetnogo poryadka s dissipatsiei”, Dokl. RAN. Mat. inform. protsessy upravl., 491:1 (2020), 95–101

[55] Shamolin M. V., “Novye sluchai odnorodnykh integriruemykh sistem s dissipatsiei na kasatelnom rassloenii dvumernogo mnogoobraziya”, Dokl. RAN. Mat. inform. protsessy upravl., 494:1 (2020), 105–111

[56] Shamolin M. V., “Novye sluchai odnorodnykh integriruemykh sistem s dissipatsiei na kasatelnom rassloenii trekhmernogo mnogoobraziya”, Dokl. RAN. Mat. inform. protsessy upravl., 495:1 (2020), 84–90

[57] Shamolin M. V., “Novye sluchai integriruemosti geodezicheskikh, potentsialnykh i dissipativnykh sistem na kasatelnom rassloenii konechnomernogo mnogoobraziya”, Dokl. RAN. Mat. inform. protsessy upravl., 500:1 (2021), 78–86

[58] Shamolin M. V., “Novye sluchai odnorodnykh integriruemykh sistem s dissipatsiei na kasatelnom rassloenii chetyrekhmernogo mnogoobraziya”, Dokl. RAN. Mat. inform. protsessy upravl., 497:1 (2021), 23–30

[59] Shamolin M. V., “Tenzornye invarianty geodezicheskikh, potentsialnykh i dissipativnykh sistem na kasatelnom rassloenii dvumernogo mnogoobraziya”, Dokl. RAN. Mat. inform. protsessy upravl., 501:1 (2021), 89–94

[60] Shamolin M. V., “Invariantnye formy ob'ema sistem s tremya stepenyami svobody s peremennoi dissipatsiei”, Dokl. RAN. Mat. inform. protsessy upravl., 507:1 (2022), 86–92

[61] Shamolin M. V., “Integriruemye odnorodnye dinamicheskie sistemy s dissipatsiei na kasatelnom rassloenii gladkogo konechnomernogo mnogoobraziya. I. Uravneniya geodezicheskikh na kasatelnom rassloenii gladkogo $n$-mernogo mnogoobraziya”, Itogi nauki tekhn. Sovr. mat. prilozh. Temat. obz., 214 (2022), 82–106

[62] Shamolin M. V., “Integriruemye odnorodnye dinamicheskie sistemy s dissipatsiei na kasatelnom rassloenii gladkogo konechnomernogo mnogoobraziya. II. Uravneniya dvizheniya na kasatelnom rassloenii k $n$-mernomu mnogoobraziyu v potentsialnom silovom pole”, Itogi nauki tekhn. Sovr. mat. prilozh. Temat. obz., 215 (2022), 81–94

[63] Shamolin M. V., “Integriruemye odnorodnye dinamicheskie sistemy s dissipatsiei na kasatelnom rassloenii gladkogo konechnomernogo mnogoobraziya. III. Uravneniya dvizheniya na kasatelnom rassloenii k $n$-mernomu mnogoobraziyu v silovom pole s peremennoi dissipatsiei”, Itogi nauki tekhn. Sovr. mat. prilozh. Temat. obz., 216 (2022), 133–152

[64] Shamolin M. V., “Integriruemye odnorodnye dinamicheskie sistemy s dissipatsiei na kasatelnom rassloenii trekhmernogo mnogoobraziya”, Itogi nauki tekhn. Sovr. mat. prilozh. Temat. obz., 205 (2022), 22–54

[65] Shamolin M. V., “Integriruemye odnorodnye dinamicheskie sistemy s dissipatsiei na kasatelnom rassloenii chetyrekhmernogo mnogoobraziya. I. Uravneniya geodezicheskikh”, Itogi nauki tekhn. Sovr. mat. prilozh. Temat. obz., 210 (2022), 77–95

[66] Shamolin M. V., “Integriruemye odnorodnye dinamicheskie sistemy s dissipatsiei na kasatelnom rassloenii chetyrekhmernogo mnogoobraziya. II. Potentsialnye silovye polya”, Itogi nauki tekhn. Sovr. mat. prilozh. Temat. obz., 211 (2022), 29–40

[67] Shamolin M. V., “Integriruemye odnorodnye dinamicheskie sistemy s dissipatsiei na kasatelnom rassloenii chetyrekhmernogo mnogoobraziya. III. Silovye polya s dissipatsiei”, Itogi nauki tekhn. Sovr. mat. prilozh. Temat. obz., 212 (2022), 120–138

[68] Shamolin M. V., “Nekotorye tenzornye invarianty geodezicheskikh, potentsialnykh i dissipativnykh sistem na kasatelnom rassloenii dvumernogo mnogoobraziya”, Itogi nauki tekhn. Sovr. mat. prilozh. Temat. obz., 209 (2022), 108–116

[69] Shamolin M. V., “Nekotorye tenzornye invarianty geodezicheskikh, potentsialnykh i dissipativnykh sistem na kasatelnom rassloenii trekhmernogo mnogoobraziya”, Itogi nauki tekhn. Sovr. mat. prilozh. Temat. obz., 210 (2022), 96–105

[70] Shamolin M. V., “Sistemy s konechnym chislom stepenei svobody s dissipatsiei: analiz i integriruemost. I. Porozhdayuschaya zadacha iz dinamiki mnogomernogo tverdogo tela, pomeschennogo v nekonservativnoe pole sil”, Itogi nauki tekhn. Sovr. mat. prilozh. Temat. obz., 211 (2022), 41–74

[71] Shamolin M. V., “Sistemy s konechnym chislom stepenei svobody s dissipatsiei: analiz i integriruemost. II. Obschii klass dinamicheskikh sistem na kasatelnom rassloenii mnogomernoi sfery”, Itogi nauki tekhn. Sovr. mat. prilozh. Temat. obz., 212 (2022), 139–148

[72] Shamolin M. V., “Sistemy s konechnym chislom stepenei svobody s dissipatsiei: analiz i integriruemost. III. Sistemy na kasatelnykh rassloeniyakh gladkikh $n$-mernykh mnogoobrazii”, Itogi nauki tekhn. Sovr. mat. prilozh. Temat. obz., 213 (2022), 96–109

[73] Shamolin M. V., “Sistemy s pyatyu stepenyami svobody s dissipatsiei: analiz i integriruemost. I. Porozhdayuschaya zadacha iz dinamiki mnogomernogo tverdogo tela, pomeschennogo v nekonservativnoe pole sil”, Itogi nauki tekhn. Sovr. mat. prilozh. Temat. obz., 208 (2022), 91–121

[74] Shamolin M. V., “Sistemy s pyatyu stepenyami svobody s dissipatsiei: analiz i integriruemost. II. Dinamicheskie sistemy na kasatelnykh rassloeniyakh”, Itogi nauki tekhn. Sovr. mat. prilozh. Temat. obz., 209 (2022), 88–107

[75] Shamolin M. V., “Sistemy s chetyrmya stepenyami svobody s dissipatsiei: analiz i integriruemost”, Itogi nauki tekhn. Sovr. mat. prilozh. Temat. obz., 205 (2022), 55–94

[76] Shamolin M. V., “Invarianty odnorodnykh dinamicheskikh sistem pyatogo poryadka s dissipatsiei”, Dokl. RAN. Mat. inform. protsessy upravl., 514:1 (2023), 98–106

[77] Shamolin M. V., “Invariantnye formy geodezicheskikh, potentsialnykh i dissipativnykh sistem na kasatelnom rassloenii konechnomernogo mnogoobraziya”, Dokl. RAN. Mat. inform. protsessy upravl., 512:1 (2023), 10–17

[78] Shamolin M. V., “Invariantnye formy ob'ema geodezicheskikh, potentsialnykh i dissipativnykh sistem na kasatelnom rassloenii chetyrekhmernogo mnogoobraziya”, Dokl. RAN. Mat. inform. protsessy upravl., 509:1 (2023), 69–76

[79] Shamolin M. V., “Invarianty geodezicheskikh, potentsialnykh i dissipativnykh sistem s tremya stepenyami svobody”, Differ. uravn., 60:3 (2024), 322–345

[80] Shamolin M. V., “Invarianty odnorodnykh dinamicheskikh sistem sedmogo poryadka s dissipatsiei”, Dokl. RAN. Mat. inform. protsessy upravl., 516:1 (2024), 65–74

[81] Shamolin M. V., “Invarianty sistem s malym chislom stepenei svobody, obladayuschikh dissipatsiei”, Vestn. Mosk. un-ta. Ser. 1. Mat. Mekh., 2024, no. 2, 3–15

[82] Polyanin A. D., Zaitsev V. F., Handbook of Ordinary Differential Equations: Exact Solutions, Methods, and Problems, Chapman and Hall, New York, 2017

[83] Poincaré H., Calcul des probabilités, Gauthier-Villars, Paris, 1912

[84] Shamolin M. V., “Some questions of the qualitative theory of ordinary differential equations and dynamics of a rigid body interacting with a medium”, J. Math. Sci., 110:2 (2002), 2528–2557

[85] Shamolin M. V., “Invariants of dynamical systems with dissipation on tangent bundles of low-dimensional manifolds”, Differential Equations, Mathematical Modeling and Computational Algorithms, ed. Vasilyev V., Springer, Cham, 2023, 167–179