关于Birkhoff逆问题中Santilli方法的研究  

On Santilli's methods in Birkhoffian inverse problem

作　　者：崔金超[1] 陈漫[2] 廖翠萃 

机构地区：[1]江南大学理学院,无锡214122 [2]北京理工大学机械与车辆学院,北京100081

出　　处：《物理学报》2018年第5期12-20,共9页Acta Physica Sinica

基　　金：国家自然科学基金(批准号:51175042;61402202;11401259)资助的课题~~

摘　　要：研究构造Birkhoff动力学函数的Santilli方法.首先,基于Cauchy-Kovalevskaya型方程解的存在性定理,采用反证法证明自治系统总有自治Birkhoff表示;其次,给出更简洁的方法证明Santilli第二方法可以被简化;找到Santilli第三方法中所隐含的一种等量关系,提出改进的Santilli第三方法,并研究该方法的MATLAB程序化计算;最后,总结全文并对结果进行讨论.In this paper, we mainly study the simplification and improvement of Santilli’s methods in Birkhoffian system, which is a more general type of basic dynamic system. The theories and methods of Birkhoffian dynamics have been used in hadron physics, quantum physics, rotational relativity theory, and fractional dynamics. As is well known, Lagrangian inverse problem, Hamiltonian inverse problem, and Birkhoffian inverse problem are the main objects of the dynamic inverse problems. The results given by Douglas（Douglas J 1941 Trans. Amer. Math. Soc. 50 71） and Havas [Havas P 1957 Nuovo Cimento Suppl. Ser. X5 363] show that only the self-adjoint Newtonian systems can be represented by Lagrange’s equations, so the Lagrangian inverse problem is not universal for a holonomic constrained mechanical system.Furthermore, from the equivalence between Lagrange’s equation and Hamilton’s equation, Hamiltonian inverse problem is not universal. A natural question is then raised： whether there exists a self-adjoint dynamical model whose inverse problem is universal for holonomic constrained mechanical systems, in the field of analytical mechanics.An in-depth study of this issue in the 1980 s by R. M. Santilli shows that a universal self-adjoint model exists for a holonomic constrained mechanic system that satisfies the basic conditions of locality, analyticity, and formality.The Birkhoff’s equation is a natural extension of the Hamilton’s equation, which shows the geometric properties of a nonconservative system as a general symplectic structure. This more general symplectic structure provides the geometry for the study of the non-conservative system preserving structure algorithms. Therefore, it is particularly important to study the problem of the Birkhoffian representation for the holonomic constrained system.For the inverse problem of Birkhoff’s dynamics, studied mainly are the condition under which the mechanical systems can be represented by Birkhoff’s equations and the construction method of Birkhoff’s f

关 键 词：Birkhoff逆问题 Birkhoff动力学函数 Santilli方法 约束力学系统 

分 类 号：O316[理学—一般力学与力学基础]