约束力学系统的联络及其运动方程的测地性质<sup>*</sup>

约束力学系统的联络及其运动方程的测地性质^*

基金项目: * 国家自然科学基金;高等学校博士点专项基金

1.
Shangqiu Teachers College, Shangqiu, Henan 476000, P.R.China;
2.
Department of Applied Mechanics, Beijing Institute of Technology, Beijing 100081, P.R.China

摘要: 用现代整体微分几何方法研究非定常约束力学系统运动方程的测地性质,得到非定常力学系统的动力学流关于1-射丛上的联络具有测地性质的充分必要条件.非定常情形下的动力学流关于无挠率的联络总具有测地性质,因此任何非定常约束力学系统在外力作用下的运动总可以表示为关于1-射丛上无挠率的动力学联络的测地运动,这与定常力学的情形有所区别.
- 1-射丛 /
- 动力学流 /
- 竖直自同态 /
- 联络 /
- 测地线
Abstract: The geodesic characteristic of equations of motion for nonautonomous constrained mechanical systems is studied in the modern setting of global differential geometry.A necessary and sufficient condition for the dynamical flow of nonautonomous mechanical system with geodesic characteristic was obtained with respect to a connection on 1-jet bundle.The dynamical flow concerning the non-autonomous case was always of geocesic characteristic with regard to torsionfree connections.Thus the motion of any nonautonomous mechanical system with constraints can be always represented by the motion along the geodesic line of torsion-connection on 1-jet bundle,which is different from the case in an autonomous mechaincal system.
- 1-jet bundle /
- dynamical flow /
- vertical endomorphism /
- connection /
- geodesic

[1]	G.Godbillion,Geometrie Differ entielle et Meca nique An alytique,Hermann,Paris (1969).
[2]	V.Ⅰ.Arnold,Ma them atical Method of Classical Mechanics,Springer-Verlag,New York (1978).
[3]	R.Abraham and J.E.Marsden,Foundations of Mechanics,2nded.,The Benjamin/ Cumming Publishing Company (1978).
[4]	梅凤翔、刘端、罗勇,《高等分析力学》,北京理工大学出版社,北京 (1991).
[5]	D.Chinea,M.deleon and J.C.Marrero,The constraint algorithm for time-dependent Lagrangians,J.Math.Phys.,35(7) (1994),3410-3438.
[6]	E.Cartan,Surles Varietes a connexion affine at la theorie de la relativite generalistee,Annales de L'Ecole Nomale Superieure,40 (1923),325-412.
[7]	E.Cartan,Surles Varietes a connexion affine at la theorie de la relativite generalistee,Annales de L'Ecole Nomale Superieure,41 (1924),1-25.M.de leon and P.R.Rodrigues,Methods of Difer ential Geom etry in Analytical Mechanics,North Holland,Amsterdam (1989).
[8]	W.Sarlet,A.Vandecasteele and F.Cantrijn,Derivations of forms along a map:the framework for time-dependent second-order equations,Diff.Geom.Appl.,(1995),171-203.
[9]	Echeverria-Enriquez,M.C.Munoz-Lecanda and N.Roman-Roy,Non-standard connections in classicalmechanics,J.Phys.A:Math Gen.,2(1995),5553-5567.
[10]	G.S.Hall and B.M.Haddow,Geometrical aspects and generalizations of Newton-Cartan mechanics,Int.J.Theor.Phys.,34(7) (1995),1093-1112.
[11]	S.Kobayashi and K.Nomizu,Foundations of Differential Geometry,J.Willey and Sons,New York (1963).