Generalized Birkhoffian realization of nonholonomic systems

Yong-Xin Guo; Chang Liu; Shi-Xing Liu

Communications in Mathematics (2010)

  • Volume: 18, Issue: 1, page 21-35
  • ISSN: 1804-1388

Abstract

top
Based on the Cauchy-Kowalevski theorem for a system of partial differential equations to be integrable, a kind of generalized Birkhoffian systems (GBSs) with local, analytic properties are put forward, whose manifold admits a presymplectic structure described by a closed 2-form which is equivalent to the self-adjointness of the GBSs. Their relations with Birkhoffian systems, generalized Hamiltonian systems are investigated in detail. Analytic, algebraic and geometric properties of GBSs are formulated, together with their integration methods induced from the Birkhoffian systems. As an important example, nonholonomic systems are reduced into GBSs, which gives a new approach to some open problems of nonholonomic mechanics.

How to cite

top

Guo, Yong-Xin, Liu, Chang, and Liu, Shi-Xing. "Generalized Birkhoffian realization of nonholonomic systems." Communications in Mathematics 18.1 (2010): 21-35. <http://eudml.org/doc/196944>.

@article{Guo2010,
abstract = {Based on the Cauchy-Kowalevski theorem for a system of partial differential equations to be integrable, a kind of generalized Birkhoffian systems (GBSs) with local, analytic properties are put forward, whose manifold admits a presymplectic structure described by a closed 2-form which is equivalent to the self-adjointness of the GBSs. Their relations with Birkhoffian systems, generalized Hamiltonian systems are investigated in detail. Analytic, algebraic and geometric properties of GBSs are formulated, together with their integration methods induced from the Birkhoffian systems. As an important example, nonholonomic systems are reduced into GBSs, which gives a new approach to some open problems of nonholonomic mechanics.},
author = {Guo, Yong-Xin, Liu, Chang, Liu, Shi-Xing},
journal = {Communications in Mathematics},
keywords = {inverse problem; self-adjointness condition},
language = {eng},
number = {1},
pages = {21-35},
publisher = {University of Ostrava},
title = {Generalized Birkhoffian realization of nonholonomic systems},
url = {http://eudml.org/doc/196944},
volume = {18},
year = {2010},
}

TY - JOUR
AU - Guo, Yong-Xin
AU - Liu, Chang
AU - Liu, Shi-Xing
TI - Generalized Birkhoffian realization of nonholonomic systems
JO - Communications in Mathematics
PY - 2010
PB - University of Ostrava
VL - 18
IS - 1
SP - 21
EP - 35
AB - Based on the Cauchy-Kowalevski theorem for a system of partial differential equations to be integrable, a kind of generalized Birkhoffian systems (GBSs) with local, analytic properties are put forward, whose manifold admits a presymplectic structure described by a closed 2-form which is equivalent to the self-adjointness of the GBSs. Their relations with Birkhoffian systems, generalized Hamiltonian systems are investigated in detail. Analytic, algebraic and geometric properties of GBSs are formulated, together with their integration methods induced from the Birkhoffian systems. As an important example, nonholonomic systems are reduced into GBSs, which gives a new approach to some open problems of nonholonomic mechanics.
LA - eng
KW - inverse problem; self-adjointness condition
UR - http://eudml.org/doc/196944
ER -

References

top
  1. Bloch, A.M., Fernandez, O.E., Mestdag, T., 10.1016/S0034-4877(09)90001-5, Rep. Math. Phys. 63 2009 225–249 (2009) Zbl1207.37045MR2519467DOI10.1016/S0034-4877(09)90001-5
  2. Bloch, A.M., Baillieul, J., Crouch, P., Marsden J., Nonholonomic Mechanics and Control, Springer, London 2003 (2003) Zbl1045.70001MR1978379
  3. Cortes, J.M., Geometric, Control and Numerical Aspects of Nonholonomic Systems, Springer, Berlin 2002 (2002) Zbl1009.70001MR1942617
  4. Guo, Y.X., Luo, S.K., Shang, M., Mei, F.X., 10.1016/S0034-4877(01)80046-X, Rep. Math. Phys. 47 2001 313–322 (2001) MR1847630DOI10.1016/S0034-4877(01)80046-X
  5. Hojman, S., Construction of genotopic transformations for first order systems of differential equations, Hadronic J. 5 1981 174–184 (1981) Zbl0515.70022MR0642608
  6. Ibort, L.A., Solano, J.M., 10.1088/0266-5611/7/5/005, Inverse Problems 7 1991 713–725 (1991) Zbl0756.34019MR1128637DOI10.1088/0266-5611/7/5/005
  7. Krupková, O., Musilová, J., 10.1016/S0034-4877(05)80028-X, Rep. Math. Phys. 55 2005 211–220 (2005) DOI10.1016/S0034-4877(05)80028-X
  8. Li, J.B., Zhao, X.H., Liu, Z.R., Theory of Generalized Hamiltonian Systems and Its Applications, Science Press of China Beijing 2007 (2007) 
  9. Liu, C., Liu, S.X., Guo, Y.X., Inverse problem for Chaplygin’s nonholonomic, Sci. Chin. G 53 2010 (to appear) (2010) 
  10. Massa, E., Pagani, E., Jet bundle geometry, dynamical connections, and the inverse problem of Lagrangian mechanics, Ann. Inst. Henri Poincaré: Phys. Theor. 61 1994 17–62 (1994) Zbl0813.70004MR1303184
  11. Mei, F.X., Shi, R.C., Zhang, Y.F., Wu, H.B., Dynamics of Birkhoffian systems, Press of Beijing Institute of Technology Beijing 1996 (in Chinese) (1996) 
  12. Morando, P., Vignolo, S., 10.1088/0305-4470/31/40/015, J. Phys. A: Math. Gen. 31 1998 8233–8245 (1998) Zbl0940.70008MR1651497DOI10.1088/0305-4470/31/40/015
  13. Santilli, R.M., Foundations of Theoretical Mechanics I, Springer-Verlag, New York 1978 (1978) Zbl0401.70015MR0514210
  14. Santilli, R.M., Foundations of Theoretical Mechanics II, Springer-Verlag, New York 1983 (1983) Zbl0536.70001MR0681293
  15. Sarlet, W., 10.1088/0305-4470/15/5/013, J. Phys. A: Math. Gen. 15 1982 1503–1517 (1982) Zbl0537.70018MR0656831DOI10.1088/0305-4470/15/5/013
  16. Sarlet, W., Cantrijn, F., Saunders, D.J., 10.1088/0305-4470/30/11/029, J. Phys. A: Math. Gen. 30 1997 4031–4052 (1997) Zbl0932.37040MR1457421DOI10.1088/0305-4470/30/11/029
  17. Sarlet, W., Cantrijn, F., Saunders, D.J., 10.1088/0305-4470/28/11/022, J. Phys. A: Math. Gen. 28 1995 3253–3268 (1995) Zbl0858.70013MR1344117DOI10.1088/0305-4470/28/11/022
  18. Sarlet, W., Thompson, G., Prince, G.E., 10.1090/S0002-9947-02-02994-X, Trans. Amer. Math. Soc. 354 2002 2897–2919 (2002) MR1895208DOI10.1090/S0002-9947-02-02994-X
  19. Saunders, D.J., Sarlet, W., Cantrijn, F., 10.1088/0305-4470/29/14/042, J. Phys. A: Math. Gen. 29 1996 4265–4274 (1996) Zbl0900.70196MR1406933DOI10.1088/0305-4470/29/14/042

NotesEmbed ?

top

You must be logged in to post comments.