关键词: Hamilton-Jacobi理论/
非保守Hamilton系统/
Hamilton正则方程
English Abstract
A kind of non-conservative Hamilton system solved by the Hamilton-Jacobi method
Wang Yong1,2,Mei Feng-Xiang1,
Xiao Jing2,
Guo Yong-Xin3,4
1.School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China;
2.School of Information Engineering, Guangdong Medical University, Dongguan 523808, China;
3.College of Physics, Liaoning University, Shenyang 110036, China;
4.Department of Medical Imaging Physics, Eastern Liaoning University, Dandong 118001, China
Fund Project:Project supported by the National Natural Science Foundation of China (Grant Nos. 11572145, 11272050, 11572034) and the Natural Science Foundation of Guangdong Province, China (Grant No. 2015A030310127).Received Date:18 August 2016
Accepted Date:03 December 2016
Published Online:05 March 2017
Abstract:The Hamilton-Jacobi equation is an important nonlinear partial differential equation. In particular, the classical Hamilton-Jacobi method is generally considered to be an important means to solve the holonomic conservative dynamics problems in classical dynamics. According to the classical Hamilton-Jacobi theory, the classical Hamilton-Jacobi equation corresponds to the canonical Hamilton equations of the holonomic conservative dynamics system. If the complete solution of the classical Hamilton-Jacobi equation can be found, the solution of the canonical Hamilton equations can be found by the algebraic method. From the point of geometry view, the essential of the Hamilton-Jacobi method is that the Hamilton-Jacobi equation promotes the vector field on the cotangent bundle T* M to a constraint submanifold of the manifold T* M R, and if the integral curve of the promoted vector field can be found, the projection of the integral curve in the cotangent bundle T* M is the solution of the Hamilton equations. According to the geometric theory of the first order partial differential equations, the Hamilton-Jacobi method may be regarded as the study of the characteristic curves which generate the integral manifolds of the Hamilton 2-form . This means that there is a duality relationship between the Hamilton-Jacobi equation and the canonical Hamilton equations. So if an action field, defined on UI (U is an open set of the configuration manifold M, IR), is a solution of the Hamilton-Jacobi equation, then there will exist a differentiable map from MR to T* MR which defines an integral submanifold for the Hamilton 2-form . Conversely, if * =0 and H1(UI)=0 (H1(UI) is the first de Rham group of U I), there will exist an action field S satisfying the Hamilton-Jacobi equation. Obviously, the above mentioned geometric theory can not only be applicable to the classical Hamilton-Jacobi equation, but also to the general Hamilton-Jacobi equation, in which some first order partial differential equations correspond to the non-conservative Hamiltonian systems. The geometry theory of the Hamilton-Jacobi method is applied to some special non-conservative Hamiltonian systems, and a new Hamilton-Jacobi method is established. The Hamilton canonical equations of the non-conservative Hamiltonian systems which are applied with non-conservative force Fi = (t)pi can be solved with the new method. If a complete solution of the corresponding Hamilton-Jacobi equation can be found, all the first integrals of the non-conservative Hamiltonian system will be found. The classical Hamilton-Jacobi method is a special case of the new Hamilton-Jacobi method. Some examples are constructed to illustrate the proposed method.
Keywords: Hamilton-Jacobi theory/
non-conservative Hamilton system/
Hamilton canonical equation
