物理学报2017，Vol.66Issue(5)：181-187,7.DOI:10.7498/aps.66.054501
一类可用Hamilton-Jacobi方法求解的非保守Hamilton系统
A kind of non-conservative Hamilton system solved by the Hamilton-Jacobi method
摘要
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 ω. 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 U × I (U is an open set of the configuration manifold M , I ?R), is a solution of the Hamilton-Jacobi equation, then there will exist a differentiable map φ from M × R to T?M × R which defines an integral submanifold for the Hamilton 2-formω. Conversely, if φ?ω = 0 and H1(U × I) = 0 (H1(U × I) 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 = μ(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.关键词
Hamilton-Jacobi理论/非保守Hamilton系统/Hamilton正则方程Key words
Hamilton-Jacobi theory/non-conservative Hamilton system/Hamilton canonical equation引用本文复制引用
王勇,梅凤翔,肖静,郭永新..一类可用Hamilton-Jacobi方法求解的非保守Hamilton系统[J].物理学报,2017,66(5):181-187,7.基金项目
国家自然科学基金(批准号:11572145, 11272050, 11572034)和广东省自然科学基金(批准号:2015A030310127)资助的课题.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). （批准号:11572145, 11272050, 11572034）
