本文選題：后牛頓 + 拉格朗日��； 參考：《南昌大學(xué)》2017年碩士論文

【摘要】：對于拉格朗日與哈密頓系統(tǒng)近年來許多科研工作者進(jìn)行了研究,這些研究工作大致都是圍繞兩者等價性進(jìn)行的,即什么時候等價什么時候不等價。因此,本文也將追隨前人的腳步繼續(xù)深入研究。根據(jù)目前研究表明,致密天體往往是強引力系統(tǒng),其研究需要借助愛因斯坦廣義相對論引力理論。通常愛因斯坦場方程沒有分析解,后牛頓近似方法便被廣泛應(yīng)用來近似求解。像在牛頓力學(xué)中那樣,拉格朗日函數(shù)與哈密頓函數(shù)仍然是后牛頓力學(xué)中兩種常用表述形式,前者最為常見。但由于后者采用正則變量而具有正則動力系統(tǒng)性質(zhì)的優(yōu)點,所以常常將前者轉(zhuǎn)換成后者來研究。毫無疑問二者在牛頓力學(xué)中無疑是等價的,并且根本不需論證,但在后牛頓力學(xué)框架內(nèi)等價與否的問題卻很難回答。事實上,分析致密雙星系統(tǒng)先從拉格朗日著手再去研究其對應(yīng)的哈密頓。通過多種的模型構(gòu)造,研究了利用勒讓德變換公式將拉格朗日轉(zhuǎn)換到哈密頓時,在耦合的情況下會產(chǎn)生一些新的哈密頓后牛頓項。理論和數(shù)值分析結(jié)合說明了幾點,第一:同等階數(shù)下拉格朗日和哈密頓的不等價性;第二:在哈密頓部分中的自旋自旋軌道耦合是引起混沌的主要原因;第三:不含自旋軌道耦合的拉格朗日系統(tǒng)甚至比含有自旋軌道耦合更容易引起混沌。對于線性動量的高階自旋軌道耦合致密雙星的動力學(xué)研究,主要是在一體哈密頓的形勢下去研究拉格朗日部分的特征。因為對于含有自旋軌道項和牛頓項的哈密頓形式通過勒讓德變換為拉格朗日時會產(chǎn)生一些高階自旋自旋耦合項,所以也要嚴(yán)格的進(jìn)行比較。這個部分主要是考慮了新產(chǎn)生的這些高階自旋自旋項會怎樣改變系統(tǒng)的動力學(xué)。同時在研究與之近似等價的哈密頓,這里所得的近似等價哈密頓可以和原先的進(jìn)行比較,并討論哈密頓的可積性。隨后再去考慮哈密頓中含有自旋自旋項的效果,所得的拉格朗日又將會發(fā)生什么變化,這也是研究的一個重點。
[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.
【學(xué)位授予單位】：南昌大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O175

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相關(guān)期刊論文 前5條

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本文編號：2008747