A kind of non-conservative Hamilton system solved by the Hamilton-Jacobi method

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.