Variational and Topological Approaches to the Three-body Problem

摘要。三体问题的变分与拓扑方法。申请人将使用几何分析和经典ode的技术来研究牛顿三体问题的几个方面。提出的主要新方向是利用借鉴几何偏微分方程的“粘接”技术来理解牛顿三体问题中的散射。一个具有负能量的无界解由两个“束缚质量”组成，它们在一个接近开普勒的轨道上相互绕转，而第三个质量则渐行渐远，接近开普勒双曲轨道。两个雅可比向量中的每一个都趋近于一个开普勒二体问题的解——一个椭圆的是束缚对，一个双曲的是连接束缚对的后退质量和质心的向量。如果解在过去和未来都是无界的，那么过去和未来就会有不同的开普勒轨道——事实上，在遥远的过去被束缚的一对轨道可能与在未来被束缚的一对轨道不同。哪个开普勒参数可以通过无界三体轨道与哪个相联系？这个散射问题是我们将要研究的中心问题之一。其他需要调查的问题遵循申请人过去发起的调查路线。我们举两个例子。平面三体解可以通过其协同序列进行部分编码。是否所有的合子序列都是可能的？初步的电脑调查显示并非如此。是不是每一个趋向于拉格朗日齐次轨道的零角动量负能量解要么就是拉格朗日齐次轨道，要么就是共合轨道？牛顿三体问题关注的是三个物体的动力学，以质点为模型，通过牛顿的1/r^2引力相互吸引。把这些天体想象成地球、月亮和太阳，或者三颗星星。“问题”不是一个单独的问题，而是关于这种群众系统的长期质的行为的大量问题的集合。如果三个体间距离始终是有限的，并且始终以某个固定距离为界，则解称为“有界”。一个核心的开放性问题是：有界解的集合有多大？在任意接近有界解的地方是否存在无界解？解决这个问题的传统方法涉及到“阿诺扩散”——一种由V.I.阿诺发现的机制，通过这种机制，看似稳定的区域和轨道变得不稳定，甚至可能最终无界。阿诺扩散方法或机制在本质上是摄动的：它试图从一个明显稳定的状态中摄动出来。作为一种替代方案，我们建议从无穷远处开始，从无穷远处扫进，看看相空间的哪些部分被扫出。“从无穷远处开始”意味着考虑散射问题——一个物体与另外两个物体距离无限远的情况。最有趣的情况是，其中两个质量保持有界，当第三个质量退去时，它们在一个接近开普勒的轨道上相互绕转。想象一下，在无限的过去以这样的方式开始，在无限的未来以这样的方式结束，但是过去和未来的束缚对可能是不同的，或者它们的开普勒椭圆具有不同的偏心率、能量等。通过这种方式，我们得到了一种从开普勒轨道（或参数）到开普勒轨道的“散射图”。这个散射图是什么样的？这种“散射问题”是我们提出的主要研究方向之一。此外，我们提出了几个问题，更符合过去的调查。这里有一个例子。“syzygy”是三个质量都在一条线上的瞬间。在这样一个瞬间，一个质量“湮没”（位于）另外两个质量之间。用发生重叠的质量来标记合轨。按外观顺序列出标记的配合物，从而将每个溶液与配合物序列相关联。在平面三体问题中，是否所有的合序都是可能的？这是一个悬而未决的问题。提议者已取得部分进展。在一名本科生研究人员的帮助下，初步的计算机调查表明，答案可能是“不”。

Abstract. VARIATIONAL AND TOPOLOGICAL APPROACHES TO THE THREE-BODYPROBLEM.The proposer will investigate several aspectsof the Newtonian three-body problems usingtechniques from geometric analysis and classical ODEs. The main new direction proposed is to understand scattering in theNewtonian three-body problem using the technique of ``gluing'' borrowedfrom geometric PDE. An unbounded solution with negative energy consist of two ``bound masses'' orbiting each otherin a nearly Keplerian orbit while the third masssails away, asymptoting to a Keplerianhypebolic orbit. Each of the two Jacobi vectors asymptotes to a solution to a Keplertwo-body problems -- an elliptic one for the bound pair,and a hyperbolic one for the vector joining the receding mass to the center of mass of the bound pair.If the solution is unbounded in both the past and future,there can, and typically will, be different Keplerian orbits in the pastandthe future -- indeed the pair which is bound in the distant pastmight be different from the pair bound in the future. Which Kepler parameters can be connected to which by an unbounded three-body orbit? This scattering problem is one of the central questions we will investigate. Other problems to be investigatedfollow past lines of investigation initiated by the proposer.We give two examples. Planar three body solutions can be partially encodedbytheir syzygy sequence. Is every syzygy sequence possible?Initial computer investigations suggest not. Is it true that every zero-angular momentum negative energysolution which tends to the Lagrange homothety orbiteither is the Lagrange homothety orbit, or suffers a syzygy? The Newtonian three-body problem concerns the dynamics ofthree bodies, modelled as point masses, attracting each other by Newton's$1/r^2$ gravitational force. Think of the bodiesas the earth, moon, and sun, or as three stars. The ``problem'' is not a single problem,but rather a large collectionof problems concering the long-term qualitative behavior of such a system of masses. A solution is called ``bounded'' if the three interbody distancesremain finite and bounded by some fixed distance for all time. A central open question is: howbig is the set of bounded solutions? Is it true that arbitrarily close to a bound solution there is anunbounded one? The traditional approach to this question has involved ``Arnol'ddiffusion''-- a mechanism discovered by V.I. Arnol'd through which seemingly stableregions and orbits become unstable and perhaps eventually unbounded.The Arnol'd diffusion method or mechanism is in essence perturbative: onetries to perturb away from an apparently stable situation.As an alternative, we propose to start at infinity and sweep in frominfinity to see what parts of phase space are swept out. ``Starting at infinity'' means thinking of the scattering problem --the situation of one body infinitely distant from the other two.The most interesting case is that in which two of the masses remain bounded, orbiting each other in a nearly Keplerian orbit as the third massrecedes.Imagine beginning in such a manner in the infinite past, and ending up insuch a manner in the infinite future, but with the past and future bound pairs perhaps different,or their Keplerian ellipse having different eccentricities, energies, etc. In this way we get a kind of ``scattering map'' fromKepler orbits(or parameters) to Kepler orbits. What does this scattering map look like? This ``scattering problem''is one of the main new directions we propose to investigate.In addition, we propose several questions more in line withour past investigations. Here is one example. A ``syzygy'' is an instantat which all three masses lie on a line. At such an instant one mass ``eclipses'' (lies between) the other two.Label the syzygies by the mass doing the eclipsing.List the labelled syzygies in order of appearance, thus associating toeach solutiona syzygy sequence. In the planar three-body problem, are all syzygy sequences possible?This is an open problem. The proposer has made partial progress. Initialcomputer investigations withthe help of an undergraduate researcher suggest that the answer might be``no''.