Abstract
This paper deals with the nonholonomic mechanical systems of Chetaev's type by use of modern differential geometric methods. Based on a precise definition of Chetaev-type constraint pfaffian systems, the differential geometric structure is given for the description of nonholonomic mechanical systems. In this framwork, the classical theory of Lagrange's equations with nonholonomic constraints is put into an invariant and coordinate free form. Furthermore, the problems of constraint imbedding and conservation laws are discussed within this framwork, and the Noether-type thereom on constraint-imbedding submanifolds is obtained.
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References
Abraham, R., and J.E. Marsden,Foundations of Mechanics (2nd ed.) Benjamin/Curmming, Reading, MA (1978).
Masden, J.E., and T.J.R. Hughes,Mathematical Foundations of Elasticity, Prentice-Hall, Inc., Englewood Cliffs, N.J. (1983).
Bleecker, D.Gauge Theory and Variational Principles. Addison-Wesley Pub. Com., Inc., Massachusetts (1981).
Hermann, R.,Geometry, Physics and Systems, Dekker, New York (1973).
Edelen, D.G.B.,Lagrangian Mechanics of Nonconservative Nonholonomic Systems, Noordhoff, Leyden (1977).
Hermann, R., The Differential geometric structure of general mechanical systems from the Lagrangian point of view.J. Math. Phys. 23 (1982), 2077–2089.
Langlois, M., Sur la Mecanique analytique du corps solie et des systemes non holonomes a liaisons du type chetaev,These de Doctorat d'Etat, Besancon, France (1982).
Ghori, Q.K. and M. Hussain, Poincaré equations for nonholonomic dynamical systems,ZAMM. 53 (1973), 391–396.
Cantrijin, F., Vector fields generating invariants for classical dissipative systems.J. Math. Phys.,23 (1982), 1589–1595.
Garia, P.L., The Poincaré-Cardan invariant in the calculus variations,Symp. Math.,14 (1974), 219–246.
Poincare, H., Sur une forme nouvelle des equations de la mecanique.Comp. Rend. Acad. Sci.,132 (1901), 360–371.
Chetave, N.G., On the equations of poincare.PMM,5 (1941), 243–252. (in Russian).
Rumyantsev, V.V., On the integral principies for nonholonomic systemPMM,46 (1982). (in Russian).
Chetaev, N.G., on Gauss principle,Izv. Fiziko-Mat. Obshch. 6 (1933), 68–71 (in Rursian)
Mei Feng Xang, Nouvelles equations du mouement des systemes mecaniques nonholonomic,These de Doctorat d'Etat, Nantes, France (1982).
Noether, E., Invariante variationsprobleme, Ges. Wiss. Goettingen,2 (1981), 235–257.
Sarlet, W. and F. Cantrijn, Generalization of Noether's tneorem in classical mechanics.SIAM Rew.,23 (1981), 467–494.
Ghori, Q.K., Conservation laws for dynamical systems in Poincare-Chetaev variables,Arch. Rat. Mech. Ana. 64 (1977), 327–337.
Djukic, Dj.S., Conservation laws in classical mechanics for quasi-coordinates,Arch. Rat. Mech. Ana. 56 (1974), 79–98.
Kozlob, V.V. and N.N. Kolesnekob, On the dynamical theorems,PMM,42 (1978), 28–33. (in Russian)
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Communicated by Gue Zhong-heng
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Shi-ying, Z. The differential geometric principle of the nonholonomic mechanical systems of Chetaev's type. Appl Math Mech 7, 903–918 (1986). https://doi.org/10.1007/BF01898132
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DOI: https://doi.org/10.1007/BF01898132