Generalized Hamiltonian Formalism For Dissipative Dynamical Systems And Its Application

The most classical mechanical systems are non-conservative, therefore it is difficult to represent these non-conservative ones as standard Hamiltonian formalism which is even dimensional or other equivalent formalisms:Lagrangian formalism or variational formal-ism of least action. Because these formalisms are the base of symplectic algorithms and modern physics. Therefore, the application scope of symplectic algorithms is restricted and rare classical mechanical system can be quantized.One can employee semi analytical methods (e.g. perturbation method, method of small parameter) to solve low dimensional dissipative dynamical systems. There exist long-term problems in long-time tracking of these systems. It is evidently difficult to ap-ply analytical methods to high dimensional dissipative dynamical systems. Therefore most researchers choose numerical methods. But the results obtained by varied numerical meth-ods are widely divergent. There exists no criteria for most of nonlinear dynamical systems. Therefore, it is difficult to choose or create a proper numerical for solving these dynamical systems. Famous scientist Fengkang had proposed and investigated this problem in the area of consertive systems, and proposed the concept of symplectic algorithms and generalized theory of constructing symplectic difference schemes, and then indicated that some ex-isting difference schemes are symplectic and quite suitable for long-time tracking. Zhong had developed this idea, and then proposed time-fem method from the variational principle of dynamics, and tried to apply this class of numerical methods to damping mechanical systems. My original aim is enable symplectic algorithms to be applied to dissipative dy-namical systems such as nonlinear rotor-dynamical systems. To that end, we investigated a class of relationship between a dissipative system and conservative ones. Based on this relationship, we attempted to replace the numerical solution of the original dissipative sys-tem with the numerical solution of'substitute'conservative systems, in other words, ap-ply symplectic methods to'substitute'conservative systems. Based on this relationship, the idea of Hamilton's equation of ideal fluid and the corresponding least action principle motivated me to represent a damping dynamical system as a generalized Hamilton's equa-tion of motion and a new type least action principle (variation principle). The symplectic method of generalized Hamilton's equation proposed by Fengkang can be applied to our generalized Hamilton's equation. The new type least action principle can be incorporated in the Feynman's theory, then can be applied to the field of quantization. Our results are as followings:1. A proposition:for any non-conservative classical mechanical system and arbitrary initial condition, there exists a conservative system in a time internval; both systems sharing one and only one common phase curve; and the value of the Hamiltonian of the conservative system is equal to the sum of the total energy of the non-conservative system on the aforementioned phase curve and a constant depending on the initial condition. Based on the structure described the proposition, a class of dissipative mechanical systems are presented as corresponding infinite-dimensional Hamilto-nian systems, by defining a new Hamiltonian quantity and introducing a new Poisson bracket. The infinite-dimensional Hamiltonian systems are analogous to the Hamil-ton's equation of ideal fluid and that of plasma. This infinite-dimensional Hamilton's equation implies a new-type least-action variational principle.2. From the viewpoint in the proposition above, Euler time-centered difference scheme was investigated. The results of the investigation show that the scheme is symplectic-preserving and energy-preserving for the associated conservative system of a damp-ing oscillator. In similar way, I modified a class of explicit symplectic algorithms, so that the class of algorithms can be devoted to damping mechanical systems.3. Replace the classical least-action variational principle with the new-type of varia-tional principle, the formulation of Feynman's propagator was modified to obtain the propagator of damping systems.In present, by our method a class of symplectic methods can be applied to simulate some nonlinear dynamical systems (e.g. rotor-dynamical systems); our quantization model can be applied to quantization of a damping particle, the result approached to the result obtained by Caldirola-Kanai method, but our result is more rational.