Studies Of The Hamilton-jacobi Method For Nonconservative Systems And Linear Homogeneous Nonholonomic Systems

In the classical analytical mechanics,the Hamilton-Jacobi method is an important means to solve the Hamilton’s canonical equations of the conservative system with holomomic con-straints.This integration method has some unique advantages.So there are many problems that can only be solved by Hamilton-Jacobi method.From the geometric interpretation of the Hamilton-Jacobi method,it can be seen that the application of the Hamilton-Jacobi method is not limited to the conservative system with holomomic constraints.Based on the modern differential geometry theory,the Hamilton-Jacobi method of nonconservative systems and linear nonholonomic constraint systems are inverstigated from the following six aspects in this dissertation.(1)A new geometric interpretation is given to the Hamilton-Jacobi method based on the Frobenius theorem.The Hamilton-Jacobi method is essentially to find a suitable mapping which can push forward a vector field Y on the cotangent bundle T*Q of a mechanics system to the integrable vector field φ*Y on a high dimensional manifold.As long as this can be done,the integral curve of the vector field Y can be obtained by pulling back the integral curve of the vector field φ*Y Y.It is also pointed out that the application of the Hamilton-Jacobi method is not only limited to the conservative system with holomomic constraints.(2)The Hamilton-Jacobi method for nonconservative systems is studied.The Hamilton-Jacobi method is presented for integrating the Hamilton’s equations of the nonconservative Hamilton system with active forces Fi=μ(t)pi.It is also proved that the nonconservative Hamilton system with active forces Ri=μ(t)pi is the only nonconservative system that can be solved by the Hamilton-Jacobi equation shown as form(?)(3)It is pointed that the integrability of the first order linear mapping is not the necessary condition for the absence of the torsion of the image space of the mapping.This means that for a linear homogeneous nonholonomic system,by the first order linear nonholonomic mapping implied all constraints,we can get its reduced Riemann configuration space.And the reduced Riemann configuration space is described with a set of quasi-coordinates.By this way,the quasi-canonization of linear homogeneous nonholonomic constraint systems can be realized.The geometric essence of the quasi-canonization is that the nonholonomic mapping on a configuration space X can induce a nonholonomic mapping on the cotangent bundle T*X.And thus we can map out a immersed subbundle with symplectic structure in the cotangent bundle T*X(4)The Hamilton-Jacobi method is proposed for linear homogeneous nonholonomic systems.First,we can realize the quasi-canonization of a linear homogeneous nonholonomic system by a suitable first order linear nonholonomic mapping implied all constraints.By this method,we can use a set of quasi-coordinates and quasi-momentums to express the motion of the system into the Hamilton’s canonical equations.From this,we can naturally extend the Hamilton-Jacobi method to linear homogeneous nonholonomic systems(5)The Vujanovic field method is improved.Similar to the Hamilton-Jacobi method,the Vujanovic field method transforms the problem of solving the particular solution of an or-dinary differential equations into the problem of finding the complete solution of a first order quasilinear partial differential equation,which is usually called the basic partial differen-tial equation.due to no strong restrictive conditions as the classic Hamilton-Jacobi method,Vujanovic field method can be easily applied to the study of nonconservative systems or nonholonomic systems.Even so,there is still a fundamental difficulty in the Vujanovic field method.That is,For most dynamical systems,it is hard to find the complete solution of the basic partial differential equation.In this dissertation,the Vujanovic field method is im-proved to a new field method.The purpose of the improved field method is to find the first integrals of the motion equations,but not the particular solution of the motion equations The improved field method no longer requires that the complete solution of the basic par-tial differential equation must be obtained,thus the application scope of the field method is expanded(6)The application of the improved field method in linear homogeneous nonholonomic systems is studied.For a linear homogeneous nonholonomic system,we first realize the geometrization of constraints by the first order linear nonholonomic mapping implied all constraints.Thus the motion equations of the system in its Riemann-Cartan space can be obtained.Then some first integrals of the system can be found by using the improved field method.