Geodesic
Topology and bifurcations in nonholonomic mechanics
Russian journal of nonlinear dynamics, Tome 11 (2015) no. 4, pp. 735-762.

Voir la notice de l'article provenant de la source Math-Net.Ru

This paper develops topological methods for qualitative analysis of the behavior of nonholonomic dynamical systems. Their application is illustrated by considering a new integrable system of nonholonomic mechanics, called a nonholonomic hinge. Although this system is nonholonomic, it can be represented in Hamiltonian form with a Lie–Poisson bracket of rank 2. This Lie–Poisson bracket is used to perform stability analysis of fixed points. In addition, all possible types of integral manifolds are found and a classification of trajectories on them is presented.
Keywords: nonholonomic hinge, topology, bifurcation diagram, tensor invariants, stability.
Mots-clés : Poisson bracket 
@article{ND_2015_11_4_a7,
     author = {I. A. Bizyaev and A. V. Bolsinov and A. V. Borisov and I. S. Mamaev},
     title = {Topology and bifurcations in nonholonomic mechanics},
     journal = {Russian journal of nonlinear dynamics},
     pages = {735--762},
     publisher = {mathdoc},
     volume = {11},
     number = {4},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ND_2015_11_4_a7/}
}
TY  - JOUR
AU  - I. A. Bizyaev
AU  - A. V. Bolsinov
AU  - A. V. Borisov
AU  - I. S. Mamaev
TI  - Topology and bifurcations in nonholonomic mechanics
JO  - Russian journal of nonlinear dynamics
PY  - 2015
SP  - 735
EP  - 762
VL  - 11
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ND_2015_11_4_a7/
LA  - ru
ID  - ND_2015_11_4_a7
ER  - 
%0 Journal Article
%A I. A. Bizyaev
%A A. V. Bolsinov
%A A. V. Borisov
%A I. S. Mamaev
%T Topology and bifurcations in nonholonomic mechanics
%J Russian journal of nonlinear dynamics
%D 2015
%P 735-762
%V 11
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ND_2015_11_4_a7/
%G ru
%F ND_2015_11_4_a7
I. A. Bizyaev; A. V. Bolsinov; A. V. Borisov; I. S. Mamaev. Topology and bifurcations in nonholonomic mechanics. Russian journal of nonlinear dynamics, Tome 11 (2015) no. 4, pp. 735-762. http://geodesic.mathdoc.fr/item/ND_2015_11_4_a7/

[1] Bizyaev I. A., Borisov A. V., Mamaev I. S., “The dynamics of nonholonomic systems consisting of a spherical shell with a moving rigid body inside”, Regul. Chaotic Dyn., 19:2 (2014), 198–213 ; Бизяев B. А., Борисов А. В., Мамаев И. С., “Динамика неголономных систем, состоящих из сферической оболочки с подвижным твердым телом внутри”, Нелинейная динамика, 9:3 (2013), 547–566 | DOI | MR | Zbl | MR

[2] Bolsinov A. V., Taimanov I. A., “Integrable geodesic flows with positive topological entropy”, Invent. Math., 140:3 (2000), 639–650 | DOI | MR | Zbl

[3] Borisov A. V., Kilin A. A., Mamaev I. S., “Hamiltonicity and integrability of the Suslov problem”, Regul. Chaotic Dyn., 16:1–2 (2011), 104–116 ; Борисов А. В., Килин А. А., Мамаев И. С., “Гамильтоновость и интегрируемость задачи Суслова”, Нелинейная динамика, 6:1 (2010), 127–142 | DOI | MR | Zbl | MR

[4] Borisov A. V., Mamaev I. S., “Conservation laws, hierarchy of dynamics and explicit integration of nonholonomic systems”, Regul. Chaotic Dyn., 13:5 (2008), 443–490 | DOI | MR | Zbl

[5] Borisov A. V., Mamaev I. S., “The rolling of a rigid body on a plane and a sphere: Hierarchy of dynamics”, Regul. Chaotic Dyn., 7:2 (2002), 177–200 | DOI | MR | Zbl

[6] Borisov A. V., Mamaev I. S., Bizyaev I. A., “The hierarchy of dynamics of a rigid body rolling without slipping and spinning on a plane and a sphere”, Regul. Chaotic Dyn., 18:3 (2013), 277–328 | DOI | MR | Zbl

[7] Borisov A. V., Mamaev I. S., Kilin A. A., “The rolling motion of a ball on a sureface: New integrals and hierarchy of dynamics”, Regul. Chaotic Dyn., 7:2 (2002), 200–219 | MR

[8] Butler L., “Integrable geodesic flows with wild first integrals: The case of two-step nilmanifolds”, Ergodic Theory Dynam. Systems, 23:3 (2003), 771–797 | DOI | MR | Zbl

[9] Fuller F. B., “The writhing number of a space curve”, Proc. Natl. Acad. Sci. USA, 68 (1971), 815–819 | DOI | MR | Zbl

[10] Bizyaev I. A., Kozlov V. V., “Odnorodnye sistemy s kvadratichnymi integralami, kvaziskobki Li – Puassona i metod Kovalevskoi”, Matem. sb., 206:12 (2015), 29–54 | DOI | MR

[11] Bolsinov A. V., Borisov A. V., Mamaev I. S., “Topologiya i ustoichivost integriruemykh sistem”, UMN, 65:2(392) (2010), 71–132 | DOI | MR | Zbl

[12] Bolsinov A. V., Izosimov A. M., Konyaev A. Yu., Oshemkov A. A., “Algebra i topologiya integriruemykh sistem: Zadachi dlya issledovaniya”, Tr. sem. po vektorn. i tenzorn. anal., 28, 2012, 119–191

[13] Bolsinov A. V., Taimanov I. A., “Integriruemye geodezicheskie potoki na nadstroikakh avtomorfizmov torov”, Dinamicheskie sistemy, avtomaty i beskonechnye gruppy, Tr. MIAN, 231, Nauka, M., 2000, 46–63 | MR | Zbl

[14] Vagner V. V., “Geometricheskaya interpretatsiya dvizheniya negolonomnykh dinamicheskikh sistem”, Tr. sem. po vektorn. i tenzorn. analizu, 5, 1941, 301–327 | MR | Zbl

[15] Kozlov V. V., Kolesnikov N. N., “O teoremakh dinamiki”, PMM, 42:1 (1978), 28–33 | MR

[16] Konyaev A. Yu., “Klassifikatsiya algebr Li s orbitami koprisoedinennogo predstavleniya obschego polozheniya razmernosti $2$”, Matem. sb., 205:1 (2014), 47–66 | DOI | MR | Zbl

[17] Suslov G. K., Teoreticheskaya mekhanika, Gostekhizdat, M.–L., 1946, 655 pp.

[18] Kharlamov A. P., Kharlamov M. P., “Negolonomnyi sharnir”, NAN Ukrainy. Mekhanika tverdogo tela, 1995, no. 27, 1–7 | MR | Zbl