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Register a new userNoncollision Singularities in a Planar Four-body Problem
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Jinxin Xue
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In this paper, we show that there is a Cantor set of initial conditions in the planar four-body problem such that all four bodies escape to infinity in a finite time, avoiding collisions. This proves the Painlev{e} conjecture for the four-body case, and thus settles the last open case of the conjecture.
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In this paper, we study a model of simplified four-body problem called planar two-center-two-body problem. In the plane, we have two fixed centers $Q_1=(-chi,0)$, $Q_2=(0,0)$ of masses 1, and two moving bodies $Q_3$ and $Q_4$ of masses $mull 1$. They interact via Newtonian potential. $Q_3$ is captured by $Q_2$, and $Q_4$ travels back and forth between two centers. Based on a model of Gerver, we prove that there is a Cantor set of initial conditions which lead to solutions of the Hamiltonian system whose velocities are accelerated to infinity within finite time avoiding all early collisions. We consider this model as a simplified model for the planar four-body problem case of the Painlev{e} conjecture.
It is well known that the linear stability of Lagrangian elliptic equilateral triangle homographic solutions in the classical planar three-body problem depends on the mass parameter $bb=27(m_1m_2+m_2m_3+m_3m_1)/(m_1+m_2+m_3)^2in [0,9]$ and the eccentricity $ein [0,1)$. We are not aware of any existing analytical method which relates the linear stability of these solutions to the two parameters directly in the full rectangle $[0,9]times [0,1)$, besides perturbation methods for $e>0$ small enough, blow-up techniques for $e$ sufficiently close to 1, and numerical studies. In this paper, we introduce a new rigorous analytical method to study the linear stability of these solutions in terms of the two parameters in the full $(bb,e)$ range $[0,9]times [0,1)$ via the $om$-index theory of symplectic paths for $om$ belonging to the unit circle of the complex plane, and the theory of linear operators. After establishing the $om$-index decreasing property of the solutions in $bb$ for fixed $ein [0,1)$, we prove the existence of three curves located from left to right in the rectangle $[0,9]times [0,1)$, among which two are -1 degeneracy curves and the third one is the right envelop curve of the $om$-degeneracy curves for $om ot=1$, and show that the linear stability pattern of such elliptic Lagrangian solutions changes if and only if the parameter $(bb,e)$ passes through each of these three curves. Interesting symmetries of these curves are also observed. The singular case when the eccentricity $e$ approaches to 1 is also analyzed in details concerning the linear stability.
The restricted planar four body problem describes the motion of a massless body under the Newtonian gravitational force of other three bodies (the primaries), of which the motion gives us general solutions of the three body problem. A trajectory is called {it oscillatory} if it goes arbitrarily faraway but returns infinitely many times to the same bounded region. We prove the existence of such type of trajectories provided the primaries evolve in suitable periodic orbits.
We introduce an algebraic method to study local stability in the Newtonian $n$-body problem when certain symmetries are present. We use representation theory of groups to simplify the calculations of certain eigenvalue problems. The method should be applicable in many cases, we give two main examples here: the square central configurations with four equal masses, and the equilateral triangular configurations with three equal masses plus an additional mass of arbitrary size at the center. We explicitly found the eigenvalues of certain 8x8 Hessians in these examples, with only some simple calculations of traces. We also studied the local stability properties of corresponding relative equilibria in the four-body problems.
For the Restricted Circular Planar 3 Body Problem, we show that there exists an open set $mathcal U$ in phase space independent of fixed measure, where the set of initial points which lead to collision is $O(mu^frac{1}{20})$ dense as $murightarrow 0$.
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