Multipurpose software system for research of mechanical systems with nonholonomic constraints

I. S. Mamaev

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 2 (2009), pp. 147-160

Résumé

We consider different mechanical systems with nonholonomic constraints; in particular, we examine the existence of tensor invariants (laws of conservation) and their connection with the behavior of a system. Considerable attention is given to the possibility of conformally Hamiltonian representation of the equations of motion, which is mainly used for the integration of the considered systems.

Détail

Keywords: nonholonomic systems, conservation laws, hierarchy of dynamics, explicit integration.
Mots-clés : implementation of constraints

Bibliographie

[1] Argatov I. I., “Usloviya ravnovesiya tverdogo tela na sherokhovatoi ploskosti pri osesimmetrichnom raspredelenii normalnykh davlenii”, Izv. AN SSSR, Mekh. tv. tela, 2005, no. 2, 15–26

[2] Borisov A. V., Mamaev I. S., Puassonovy struktury i algebry Li v gamiltonovoi mekhanike, Izd-vo “RKhD”, M., Izhevsk, 1999, 464 pp. | MR | Zbl

[3] Borisov A. V., Mamaev I. S., “Shar Chaplygina, zadacha Suslova i zadacha Veselovoi. Integriruemost i realizatsiya svyazei”, Negolonomnye dinamicheskie sistemy, Izd-vo “RKhD”, IKI, M., Izhevsk, 2002, 118–130 | MR

[4] Borisov A. V., Mamaev I. S., “Integriruemaya sistema s neintegriruemoi svyazyu”, Mat. zam., 80:1 (2006), 131–134 | DOI | MR | Zbl

[5] Borisov A. V., Mamaev I. S., “Gamiltonovost zadachi Chaplygina o kachenii shara”, Mat. zam., 70:5 (2001), 793–795 | DOI | MR | Zbl

[6] Borisov A. V., Mamaev I. S., “Izomorfizm i gamiltonovo predstavlenie nekotorykh negolonomnykh sistem”, Sib. mat. zhur., 48:1 (2007), 33–45, arXiv: nlin.SI/0509036 | MR | Zbl

[7] Borisov A. V., Mamaev I. S., Marikhin V. G., “Yavnoe integrirovanie odnoi negolonomnoi zadachi”, Doklady RAN, 2008 (to appear)

[8] Borisov A. V., Fedorov Yu. N., “O dvukh vidoizmenennykh integriruemykh zadachakh dinamiki”, Vestn. MGU, ser. mat. mekh., 1995, no. 6, 102–105 | MR | Zbl

[9] Veselov A. P., Veselova L. E., “Integriruemye negolonomnye sistemy na gruppakh Li”, Mat. zametki, 44:5 (1988), 604–619 | MR

[10] Zhuravlev V. F., “O modeli sukhogo treniya v zadachakh dinamiki tverdykh tel”, Uspekhi mekhaniki, 2005, no. 3, 58–76

[11] Kozlov V. V., “K teorii integrirovaniya uravnenii negolonomnoi mekhaniki”, Uspekhi mekhaniki, 8:3 (1985), 85–101 ; Kozlov V. V., “On the Integration Theory of Equations of Nonholonomic Mechanics”, Regul. Chaotic Dyn., 7:2 (2002), 191–176 | MR | DOI | MR

[12] Kornfeld I. P., Sinai Ya. G., Fomin S. V., Ergodicheskaya teoriya, Nauka, M., 1980 | MR | Zbl

[13] Marikhin V. G., Sokolov V. V., “O parakh kommutiruyuschikh gamiltonianov, kvadratichnykh po impulsam”, TMF, 149:2 (2006), 147–160 | DOI | MR | Zbl

[14] Markeev A. P., “Ob integriruemosti zadachi o kachenii shara s mnogosvyaznoi polostyu, zapolnennoi idealnoi zhidkostyu”, Izv. AN SSSR, Mekh. tv. tela, 21:1 (1986), 64–65

[15] Puankare A., O krivykh, opredelyaemykh differentsialnymi uravneniyami, Izd-vo inostr. lit., M., 1947

[16] Suslov G. K., Teoreticheskaya mekhanika, Gostekhizdat, M., L., 1946, Utochn. reprinty

[17] Tatarinov Ya. V., “Razdelyayuschie peremennye i novye topologicheskie yavleniya v golonomnykh i negolonomnykh sistemakh”, Trudy seminara po vektorn. i tenzorn. analizu, 23, MGU, 1988, 160–174 | MR

[18] Chaplygin, S. A., “O katanii shara po gorizontalnoi ploskosti”, Sobr. soch., v. 1, OGIZ, M., L., 1948, 76–101

[19] Yaroschuk V. A., “Novye sluchai suschestvovaniya integralnogo invarianta v zadache o kachenii tverdogo tela bez proskalzyvaniya po nepodvizhnoi poverkhnosti”, Vestnik MGU, ser. mat. mekh., 1992, no. 6, 26–30

[20] Borisov A. V. and Mamaev I. S., “The Rolling of Rigid Body on a Plane and Sphere. Hierarchy of Dynamics”, Regul. Chaotic Dyn., 7:1 (2002), 177–200 | DOI | MR | Zbl

[21] Borisov A. V. and Mamaev I. S., “Rolling of a Non-homogeneous Ball Over a Sphere Without Slipping and Twisting”, Regul. Chaotic Dyn., 12:2 (2007), 153–159 | DOI | MR | Zbl

[22] Chow W. L., “Über Systeme von linearen partiellen Differential Gleichungen erster Ordnung”, Math. Ann., 117 (1939), 98–105 | DOI | MR

[23] Contensou P., “Couplage entre frottement de glissement et frottement de pivotement dans la théorie de la toupie”, Kreiselprobleme Gydrodynamics: IUTAM Symp. Celerina, Springer, Berlin, 1963, 201–216 ; “Svyaz mezhdu treniem skolzheniya i treniem vercheniya i ee uchet v teorii volchka”, V kn.: Problemy giroskopii, Mir, M., 1967, 60–77 | DOI

[24] Ferrers N. M., “Extension of Lagrange's Equations”, Quart. J. Pure Appl. Math., 12:45 (1872), 1–5

[25] Kilin A. A., “The dynamics of Chaplygin ball: the qualitative and computer analisis”, Regul. Chaotic Dyn., 6:3 (2001), 291–306 | DOI | MR | Zbl

[26] Schneider D. A., “Non-holonomic Euler–Poincaré Equations and Stability in Chaplygin's Sphere”, Dyn. Sys., 17:2 (2002), 87–130 | DOI | MR | Zbl