Variational Structure of Collisions in the Three-Body Problem

三体问题中碰撞的变分结构

• 批准号: 0072336
• 负责人: Richard Montgomery
• 金额: $ 14.31万
• 依托单位: University of California-Santa Cruz
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072336&HistoricalAwards=false
• 关键词: Variational; Structure; Collisions; Three; Body

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.

摘要奖：DMS-0072336首席研究员：理查德·W·蒙哥马利首席研究员将结合变分的直接方法、对“形状空间”的详细知识以及对物体近碰撞的作用函数的仔细研究，寻找牛顿N体问题的新解决方案。我们所说的‘’形空间‘’是指n边形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有效性，为三体问题提供了一个迄今未知的轨道。在我们的新轨道上，所有三个质量都围绕着平面上的八字形曲线相互追逐。我们的轨道是动态的(实际上是KAM)稳定的。成功应用该方法需要克服的主要技术困难是避免群众之间的碰撞。与强力势问题中的作用量不同，牛顿势作用量允许有碰撞的有限作用解。人们对有碰撞的极小化器知之甚少。具体地说，它们不需要在任何不同意义上被正规化。提出者将把重点放在碰撞上。如果一个极小化动作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正则化的？有没有爆炸技术可以让我们更好地理解这种倾向于碰撞的序列？这些都是我们将要考虑的问题。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，法国数学家庞卡雷取得了根本性的进步。他指出，在三体问题中存在混沌，而在两体问题中，运动是非常规则的(与地球围绕太阳的运动非常接近)。他还指出了周期轨道对这个问题的核心重要性。周期轨道是质量无限期地重复相同模式的运动，就像一个点绕着一个圆转。我们建议寻找三体问题和N体(N是四、五、六、…)的新的周期解。通过使用多种方法的组合来解决问题。这些方法本身并不新鲜，但它们的结合是新的。这种方法已经在一个例子中被证明是成功的--通过产生一种新的解决方案，其中三个相等的质量绕着八字曲线相互追逐，永远不会相互追赶。我们的工作可能会在理解N-体问题方面取得进一步的重大进展。这些技术在其他动态情况下可能被证明是有用的。我们的轨道有可能存在于宇宙中的某个地方，或者有朝一日被用于太空任务。