The method of nonholonomic mapping is utilized to construct a Riemann-Cartan space embedded into a known Riemann-Cartan space,which includes two special cases that a Weitzenbck space and a Riemann-Cartan space are r...The method of nonholonomic mapping is utilized to construct a Riemann-Cartan space embedded into a known Riemann-Cartan space,which includes two special cases that a Weitzenbck space and a Riemann-Cartan space are respectively embedded into a Euclidean space and a Riemann space.By means of this mapping theory,the nonholonomic corresponding relation between the autoparallels of two Riemman-Cartan spaces is investigated.In particular,an autoparallel in a Riemann-Cartan space can be mapped into a geodesic line in a Riemann space and an autoparallel in Weitzenbck space be mapped into a geodesic line in Euclidean space.Based on the Lagrange-d'Alembert principle,the equations of motion for dynamical systems in Riemman-Cartan space should be autoparallel equations of the space.As applications,the problem of autoparallel motion of spinless particles,Chaplygin's nonholonomic systems and a rigid body rotating with a fixed point are investigated in space with torsion.展开更多
The geometric formulation of motion of the first-order linear homogenous scleronomous nonholonomic system subjected to active forces is studied with the non- holonomic mapping theory. The quasi-Newton law, the quasi-m...The geometric formulation of motion of the first-order linear homogenous scleronomous nonholonomic system subjected to active forces is studied with the non- holonomic mapping theory. The quasi-Newton law, the quasi-momentum theorem, and the second kind Lagrange equation of dynamical systems are obtained in the Riemann- Cartan configuration spaces. By the nonholonomic mapping, a Euclidean configuration space or a Riemann configuration space of a dynamical system can be mapped into a Riemann-Cartan configuration space with torsion. The differential equations of motion of the dynamical system can be obtained in its Riemann-Cartan configuration space by the quasi-Newton law or the quasi-momentum theorem. For a constrained system~ the differential equations of motion in its Riemann-Cartan configuration space may be sim- pler than the equations in its Euclidean configuration space or its Riemann configuration space. Therefore, the nonholonomic mapping theory can solve some constrained prob- lems, which are difficult to be solved by the traditional analytical mechanics method. Three examples are given to illustrate the effectiveness of the method.展开更多
基金supported by the National Natural Science Foundation of China (Grant Nos 10932002, 10872084 and 10472040)the Outstanding Young Talents Training Fund of Liaoning Province of China (Grant No 3040005)+2 种基金the Research Program of Higher Education of Liaoning Province of China (Grant No 2008S098)the Program Supporting Elitists of Higher Education of Liaoning Province of China (Grant No 2008RC20)the Program of Constructing Liaoning Provincial Key Laboratory of China (Grant No 2008403009)
文摘The method of nonholonomic mapping is utilized to construct a Riemann-Cartan space embedded into a known Riemann-Cartan space,which includes two special cases that a Weitzenbck space and a Riemann-Cartan space are respectively embedded into a Euclidean space and a Riemann space.By means of this mapping theory,the nonholonomic corresponding relation between the autoparallels of two Riemman-Cartan spaces is investigated.In particular,an autoparallel in a Riemann-Cartan space can be mapped into a geodesic line in a Riemann space and an autoparallel in Weitzenbck space be mapped into a geodesic line in Euclidean space.Based on the Lagrange-d'Alembert principle,the equations of motion for dynamical systems in Riemman-Cartan space should be autoparallel equations of the space.As applications,the problem of autoparallel motion of spinless particles,Chaplygin's nonholonomic systems and a rigid body rotating with a fixed point are investigated in space with torsion.
基金Project supported by the National Natural Science Foundation of China(Nos.11772144,11572145,11472124,11572034,and 11202090)the Science and Technology Research Project of Liaoning Province(No.L2013005)+1 种基金the China Postdoctoral Science Foundation(No.2014M560203)the Natural Science Foundation of Guangdong Provience(No.2015A030310127)
文摘The geometric formulation of motion of the first-order linear homogenous scleronomous nonholonomic system subjected to active forces is studied with the non- holonomic mapping theory. The quasi-Newton law, the quasi-momentum theorem, and the second kind Lagrange equation of dynamical systems are obtained in the Riemann- Cartan configuration spaces. By the nonholonomic mapping, a Euclidean configuration space or a Riemann configuration space of a dynamical system can be mapped into a Riemann-Cartan configuration space with torsion. The differential equations of motion of the dynamical system can be obtained in its Riemann-Cartan configuration space by the quasi-Newton law or the quasi-momentum theorem. For a constrained system~ the differential equations of motion in its Riemann-Cartan configuration space may be sim- pler than the equations in its Euclidean configuration space or its Riemann configuration space. Therefore, the nonholonomic mapping theory can solve some constrained prob- lems, which are difficult to be solved by the traditional analytical mechanics method. Three examples are given to illustrate the effectiveness of the method.
