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Register a new userNon-Collision singularities in the Planar two-Center-two-Body problem
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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.
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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.
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 $murightarrow 0$.
Given a set $S$ of $n$ points in the Euclidean plane, the two-center problem is to find two congruent disks of smallest radius whose union covers all points of $S$. Previously, Eppstein [SODA97] gave a randomized algorithm of $O(nlog^2n)$ expected time and Chan [CGTA99] presented a deterministic algorithm of $O(nlog^2 nlog^2log n)$ time. In this paper, we propose an $O(nlog^2 n)$ time deterministic algorithm, which improves Chans deterministic algorithm and matches the randomized bound of Eppstein. If $S$ is in convex position, then we solve the problem in $O(nlog nloglog n)$ deterministic time. Our results rely on new techniques for dynamically maintaining circular hulls under point insertions and deletions, which are of independent interest.
The quantum problem of an electron moving in a plane under the field created by two Coulombian centers admits simple analytical solutions for some particular inter-center distances. These elementary eigenfunctions, akin to those found by Demkov for the analogous three dimensional problem, are calculated using the framework of quasi-exact solvability of a pair of entangled ODEs descendants from the Heun equation. A different but interesting situation arises when the two centers have the same strength. In this case completely elementary solutions do not exist.
We outline a new method suggested by Conway (2016) for solving the two-body problem for solid bodies of spheroidal or ellipsoidal shape. The method is based on integrating the gravitational potential of one body over the surface of the other body. When the gravitational potential can be analytically expressed (as for spheroids or ellipsoids), the gravitational force and mutual gravitational potential can be formulated as a surface integral instead of a volume integral, and solved numerically. If the two bodies are infinitely thin disks, the surface integral has an analytical solution. The method is exact as the force and mutual potential appear in closed-form expressions, and does not involve series expansions with subsequent truncation errors. In order to test the method, we solve the equations of motion in an inertial frame, and run simulations with two spheroids and two infinitely thin disks, restricted to torque-free planar motion. The resulting trajectories display precession patterns typical for non-Keplerian potentials. We follow the conservation of energy and orbital angular momentum, and also investigate how the spheroid model approaches the two cases where the surface integral can be solved analytically, i.e. for point masses and infinitely thin disks.
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