Generalized Hamiltonian Formalism for Dissipative Dynamical Systems and Its Application
|Keywords||Hamilton Dissipation Damp Symplectic numerical integration methods Quantization|
Most of the studies in classical mechanics system is not a conservative system, so it is difficult to represent such systems as the classic form of Hamiltonian mechanics (even-dimensional) and with this the equivalent Lagrangian mechanics shaped form or variational principle of least action form. Because these types of mathematical methods in the form of numerical symplectic basis and foundation of modern physics, so greatly restricted symplectic dissipative systems in the field of numerical simulation of the application as well as the quantum dissipative systems, etc. applications in the field of theoretical physics. Dissipative dynamical systems for a long time tracking problem is nonlinear mechanics research areas currently one of the difficulties. For low-dimensional dissipative dynamical systems, you can use a variety of semi-analytical methods (small parameter method, perturbation method) to solve. Even so, for the long track, there are also the so-called long-term question (accumulation of errors by the method itself causes). For high-dimensional dissipative dynamical systems, direct application of analytical methods is obviously very difficult. So much for solving such problems by numerical methods. However, different numerical methods for solving large deviation may be the result of, or even far, but most of the problems is the lack of judgment of its algorithm reference standard deviation. So for the selection or creation of such problems numerical integration methods recognized practice to become a problem. China's famous scholar Mr. Feng Kang proposed and studied in the field of conservative systems such problems, given symplectic representation of ideas and systems symplectic difference schemes constructed general approach, pointing out that the original differential format suitable for long time tracking format. Mr. Zhong Wanxie developed this idea further proposed time integration finite element and fine ideas, and dissipative dynamical systems were introduced symplectic try. The initial aim of this paper is to analyze the stability of the rotor and other dissipative dynamics problems using symplectic numerical integration methods (or rather the idea of ??using the symplectic algorithm to find the right). To achieve this study dissipative and conservative systems of a special relationship, on the basis of conservative systems with the corresponding numerical solution to replace the original dissipative systems, upcoming symplectic numerical methods applied to solve the corresponding conservative system to get to be studied Numerical Solution of the system. In this relationship, based on the reference fluid mechanics equations and generalized Hamiltonian variational principle of least action, the dissipative system representation into a infinite dimensional generalized Hamiltonian system, and accordingly brought a new variational principle of least action. You can Pingkang Wen Xian symplectic generalized Hamiltonian systems thinking applied to solve this particular infinite-dimensional Hamiltonian systems. The variational principle of least action, can be combined path integral form of quantum mechanics applied to the field of quantum mechanics. The main innovation of the above work can be summarized as follows: 1. Discovered dissipative dynamical systems and a conservative dynamical systems theory coincides with the curve: For a dissipative system and its mechanics an initial condition, corresponding to different time zones must exist a family conservative mechanical system, the family of conservative and dissipative mechanical system mechanical system has one and only one common phase curve; This family of conservative Hamiltonian of the system is the aforementioned total energy dissipation of mechanical systems. For non-conservative vibration problems, this conservative system is a nonlinear conservative dynamical systems, in which the conservative forces in a certain initial conditions and non-conservative oscillators damping force and resilience of and are equal, then its phase space necessarily the same trajectory. On this basis, the introduction of infinite dimensional Hamiltonian format to represent dissipative generalized mechanical system, in which you define a new Hamiltonian, and the introduction of a new Poisson brackets, this format is similar to the problem and represents an ideal fluid plasma Generalized Hamiltonian format. Dissipative dynamical systems where the phase space as a special fluid (no internal pressure), the initial conditions as a matter coordinates, the trajectories coincide conservative dynamical systems as Hamiltonian Hamiltonian density. Corresponds to the classical Hamiltonian variational principle, this is equivalent to a generalized Hamiltonian form new variational principle. In this variational principle in the action as a region of phase space in the role of all infinitesimal amounts. 2 From a starting point of this innovative study damped vibration problems central difference scheme, find central difference scheme corresponding to the original system for conservative nonlinear conservative dynamical system family is not only Paul but Paul Sheen's gross energy. Accordingly the existing explicit symplectic numerical integration methods to transform, we get damped vibration problems of a class of explicit symplectic numerical integration methods. And in this generalized Hamiltonian format based on the use of Feng Kang Xin generalized Hamiltonian structure of the numerical integration method, has also been the explicit symplectic numerical integration methods. 3 using the new variational principle instead of the classical variational principle, modify Feynman path integral theory, get new damped particle quantum propagator formula. The theoretical current application can be a class of explicit symplectic numerical integration method is used as in the non-linear motion of the rotor under the force of the numerical simulation; theory can be extended to this article damping particle quantum mechanics quantum fields, has been similar to the Classic Caldirola-Kanai method results, but it seems more reasonable.