后牛顿拉格朗日和哈密顿系统的动力学比较

发布时间：2018-06-12 06:40

本文选题：后牛顿 + 拉格朗日　；参考：《南昌大学》2017年硕士论文

【摘要】：对于拉格朗日与哈密顿系统近年来许多科研工作者进行了研究,这些研究工作大致都是围绕两者等价性进行的,即什么时候等价什么时候不等价。因此,本文也将追随前人的脚步继续深入研究。根据目前研究表明,致密天体往往是强引力系统,其研究需要借助爱因斯坦广义相对论引力理论。通常爱因斯坦场方程没有分析解,后牛顿近似方法便被广泛应用来近似求解。像在牛顿力学中那样,拉格朗日函数与哈密顿函数仍然是后牛顿力学中两种常用表述形式,前者最为常见。但由于后者采用正则变量而具有正则动力系统性质的优点,所以常常将前者转换成后者来研究。毫无疑问二者在牛顿力学中无疑是等价的,并且根本不需论证,但在后牛顿力学框架内等价与否的问题却很难回答。事实上,分析致密双星系统先从拉格朗日着手再去研究其对应的哈密顿。通过多种的模型构造,研究了利用勒让德变换公式将拉格朗日转换到哈密顿时,在耦合的情况下会产生一些新的哈密顿后牛顿项。理论和数值分析结合说明了几点,第一:同等阶数下拉格朗日和哈密顿的不等价性;第二:在哈密顿部分中的自旋自旋轨道耦合是引起混沌的主要原因;第三:不含自旋轨道耦合的拉格朗日系统甚至比含有自旋轨道耦合更容易引起混沌。对于线性动量的高阶自旋轨道耦合致密双星的动力学研究,主要是在一体哈密顿的形势下去研究拉格朗日部分的特征。因为对于含有自旋轨道项和牛顿项的哈密顿形式通过勒让德变换为拉格朗日时会产生一些高阶自旋自旋耦合项,所以也要严格的进行比较。这个部分主要是考虑了新产生的这些高阶自旋自旋项会怎样改变系统的动力学。同时在研究与之近似等价的哈密顿,这里所得的近似等价哈密顿可以和原先的进行比较,并讨论哈密顿的可积性。随后再去考虑哈密顿中含有自旋自旋项的效果,所得的拉格朗日又将会发生什么变化,这也是研究的一个重点。
[Abstract]:In recent years, many researchers have studied Lagrangian and Hamiltonian systems. These researches focus on the equivalence of the two systems, that is, when is equivalent and when is not equivalent. Therefore, this paper will follow the footsteps of the predecessors to continue the in-depth study. According to the present studies, dense celestial bodies are often strong gravitational systems, which need to be studied by Einstein's theory of general relativistic gravity. In general, the postNewton approximation method is widely used to solve Einstein field equation without analytical solution. As in Newtonian mechanics, Lagrangian function and Hamiltonian function are still two common expressions in post-Newtonian mechanics, the former is the most common. However, the former is often converted to the latter because the latter has the advantage of the property of the regular dynamical system. There is no doubt that the two are equivalent in Newtonian mechanics and do not need to be demonstrated at all, but the question of equivalence in the framework of post-Newtonian mechanics is difficult to answer. In fact, the analysis of dense binary systems begins with Lagrange and then studies its corresponding Hamiltonian. In this paper, by using Legendre transformation formula to convert Lagrangian to Hamiltonian, some new Newtonian terms after Hamiltonian are obtained in the case of coupling. The theory and numerical analysis show that the first is the nonequivalence of Lagrange and Hamiltonian under the same order, the second is the spin-orbit coupling in the Hamiltonian part is the main cause of chaos. Third, it is easier to cause chaos in Lagrangian systems without spin-orbit coupling than in spin-orbit coupling. The dynamical study of the high order spin orbit coupled dense binary stars with linear momentum is mainly to study the characteristics of the Lagrangian part in the case of an integral Hamiltonian. Because the Hamiltonian form with spin orbital term and Newton term will produce some higher order spin-coupling terms by Legendre transformation to Lagrange, it is also necessary to make strict comparison. This part mainly considers how these new higher order spin terms change the dynamics of the system. At the same time, we study the approximate equivalent Hamiltonian, which can be compared with the original one, and discuss the integrability of Hamiltonian. It is also an important point to consider the effect of the spin term in Hamiltonian and what will happen to Lagrange.
【学位授予单位】：南昌大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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