Variational Structure of Collisions in the Three-Body Problem

摘要：dms -0072336首席研究员：Richard W. montgomery首席研究员将结合直接变分法、“形状空间”的详细知识和对物体近碰撞作用函数的仔细研究，寻找牛顿n体问题的新解。我们所说的“形状空间”是指n个粒子的相似类或同余类的空间。在最近与Alain Chenciner的合作中，这种三管齐下的方法证明了它的实用性，为三体问题提供了一个迄今为止未知的轨道。在我们的新轨道上，这三个质量在平面上绕着同样的8字形曲线相互追逐。我们的轨道是动态（实际上是KAM）稳定的。要成功地应用这种方法，主要的技术困难是避免群众之间的碰撞。与具有强力势的问题中的作用不同，具有牛顿势的作用允许具有碰撞的有限作用解。我们对碰撞最小化知之甚少。特别是，它们不需要在任何不同的意义上进行正则化。提议者将关注碰撞。如果一个动作最小化序列趋向于一条有碰撞的曲线，在什么情况下这些碰撞是列维-奇维塔正则化的？有没有爆破技术可以让我们更好地理解这种趋向于碰撞的序列？这些是我们将考虑的一些问题。三体问题是根据牛顿物理定律理解三种质量（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，可以追溯到牛顿时代。大约100年前，法国数学家庞加莱取得了根本性的进展。他证明了混沌存在于三体问题中，而不存在于两体问题中，后者的运动是非常规则的（与地球围绕太阳的运动很接近）。他还指出了周期轨道对这个问题的核心重要性。周期轨道是质量的运动，它无限期地重复相同的模式，就像一个点绕着一个圆转一样。我们提出了一种结合方法来寻找三体问题和N体（N = 4,5,6，…）问题的新周期解。这些方法本身并不新鲜，但它们的组合却是新鲜的。这种方法在一个实例中已经被证明是成功的——它产生了一个新的解决方案，在这个解决方案中，三个等质量的物体在一个8字形曲线上互相追逐，从来没有抓住过对方。我们的工作可能会导致对n体问题理解的进一步重大进展。这些技术可能在其他动态情况下是有用的。我们的轨道有可能在宇宙的某个地方被发现，或者有一天被用于太空任务。

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.