Variational Structure of Collisions in the Three-Body Problem

摘要奖：DMS-0072336主要研究者：Richard W.蒙哥马利首席研究员将寻找新的解决方案，牛顿N体问题的组合使用的直接方法的变分法，详细的知识“形状空间”，并仔细调查的行动功能近碰撞的机构。 我们所说的“形空间”指的是N-类的相似类或同余类的空间。 在最近与Alain Chenciner的联合工作中，这种三管齐下的方法证明了它的实用性，为三体问题产生了一个迄今未知的轨道。 在我们的新轨道中，所有三个质量都围绕着平面上相同的8字形曲线相互追逐。我们的轨道原来是动态（实际上是KAM）稳定的。 要成功地应用这种方法，首先要克服的技术困难是避免物体之间的碰撞。 与具有强力势的问题中的作用量不同，具有牛顿势的作用量允许有碰撞的有限作用量解。 人们对具有碰撞的极小化器知之甚少。特别是，它们不需要在任何不同的意义上被正规化。 提议者将重点放在碰撞上。 如果一个最小作用序列趋向于一条有碰撞的曲线，那么在什么情况下这些碰撞是Levi-Civita正则化的？ 有没有爆破技术能让我们更好地理解这种趋向于碰撞的序列？ 这些是我们将要考虑的一些问题。三体问题是根据牛顿物理定律理解三个质量（行星、恒星、卫星）相互吸引的长期行为的问题。 这是数学中最古老的问题之一，可以追溯到牛顿。 大约100年前，法国数学家庞加莱取得了根本性的进展。他表明，混乱存在于三体问题，而不是两体问题，在那里的运动是非常有规律的（以及近似的地球围绕太阳）。 他还指出了周期轨道对这个问题的重要性。周期轨道是物体的运动，它像一个点绕一个圆一样，不确定地重复着同样的模式。 我们提出了寻找三体问题和N体问题（N = 4，5，6，.） 通过使用组合的方法的问题。这些方法本身并不新鲜，但它们的组合却是。 这种方法已经被证明是成功的一个例子--通过产生一个新的解决方案，其中三个相等的质量围绕一个8字曲线追逐，从不互相追赶。 我们的工作可能会导致进一步的重大进展的理解N体问题。该技术可能被证明是有用的，在其他动态的情况。 有一种可能性，我们的轨道可能会被发现存在于宇宙的某个地方，或者有一天用于太空任务。

AbstractAward: DMS-0072336Principal Investigator: Richard W. MontgomeryThe principal investigator will search for new solutions to theNewtonian N-body problem using a combination of the direct methodof the calculus of variations, a detailed knowledge of ``shapespace'', and a careful investigation of the action functionalnear collisions of the bodies. By the ``shape space'' we meanthe space of either similarity classes or congruence classes ofN-gons. In recent joint work with Alain Chenciner, thisthree-pronged approach proved its utility by yielding a hithertounknown orbit for the three-body problem. In our new orbit allthree masses chase each other around the same figure eight shapedcurve in the plane. Our orbit turns out to be dynamically(actually KAM) stable. The chief technical difficulty to beovercome in successfully applying the method is that of avoidingcollisions between the masses. Unlike the action in problemswith strong-force potentials, the action with the Newtonianpotential admits finite-action solutions with collision. Oneknows very little about minimizers with collision. In particularthey need not be regularized in any of the various senses. Theproposer will focus on the collisions. If an action minimizingsequence tends to a curve with collisions, under whatcircumstance are those collisions Levi-Civita regularized? Arethere blow-up techniques which will enable us to betterunderstand such sequences tending toward collision? These aresome of the questions we will consider.The three-body problem is the problem of understanding the longterm behaviour of three masses (planets, stars, satellites)attracting each other according to Newton's laws of physics. Itis one of the oldest problems in mathematics,dating back toNewton. About 100 years ago the French mathematician Poincaremade fundamental progress. He showed that chaos exists in thethree-body problem, in contrast to the the two-body problem,where the motions are very regular (and well-approximated by thatof the earth around the sun). He also pointed out the centralimportance of periodic orbits to the problem. Periodic orbits aremotions of the masses which repeat the same pattern indefinitelylike a point going around a circle. We propose to find newperiodic solutions to the three-body problem and the N-body (N isfour, five, six,...) problem by using a combination ofmethods. The methods themselves are not new, but theircombination is. This approach has already proved successful inone instance -- by yielding a new solution in which three equalmasses chase each around a figure eight curve, never catchingeach other. Our work could lead to further significant advancesin the understanding of the N-body problem. The techniques mayprove to be useful in other dynamical situations. There is somepossibility that our orbits might be found to exist somewhere inthe universe, or used in space missions someday.