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Register a new userNotes on spatial twisted central configurations for $2N$-body problem
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We study the spatial central configuration formed by two twisted regular $N$-polygons. For any twist angle $theta$ and any ratio of the masses $b$ in the two regular $N$-polygons, we prove that the sizes of the two regular $N$-polygons must be equal.
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For the Newtonian (gravitational) $n$-body problem in the Euclidean $d$-dimensional space, the simplest possible solutions are provided by those rigid motions (homographic solutions) in which each body moves along a Keplerian orbit and the configuration of the $n$-body is a constant up to rotations and scalings named textit{central configuration}. For $dleq 3$, the only possible homographic motions are those given by central configurations. For $d geq 4$ instead, new possibilities arise due to the higher complexity of the orthogonal group $O(d)$, as observed by Albouy and Chenciner. For instance, in $mathbb R^4$ it is possible to rotate in two mutually orthogonal planes with different angular velocities. This produces a new balance between gravitational forces and centrifugal forces providing new periodic and quasi-periodic motions. So, for $dgeq 4$ there is a wider class of $S$-textit{balanced configurations} (containing the central ones) providing simple solutions of the $n$-body problem, which can be characterized as well through critical point theory. In this paper, we first provide a lower bound on the number of balanced (non-central) configurations in $mathbb R^d$, for arbitrary $dgeq 4$, and establish a version of the $45^circ$-theorem for balanced configurations, thus answering some questions raised by Moeckel. Also, a careful study of the asymptotics of the coefficients of the Poincare polynomial of the collision free configuration sphere will enable us to derive some rather unexpected qualitative consequences on the count of $S$-balanced configurations. In the last part of the paper, we focus on the case $d=4$ and provide a lower bound on the number of periodic and quasi-periodic motions of the gravitational $n$-body problem which improves a previous celebrated result of McCord.
For the gravitational $n$-body problem, the simplest motions are provided by those rigid motions in which each body moves along a Keplerian orbit and the shape of the system is a constant (up to rotations and scalings) configuration featuring suitable properties. While in dimension $d leq 3$ the configuration must be central, in dimension $d geq 4$ new possibilities arise due to the complexity of the orthogonal group, and indeed there is a wider class of $S$-balanced configurations, containing central ones, which yield simple solutions of the $n$-body problem. Starting from recent results of the first and third authors, we study the existence of continua of bifurcations branching from a trivial branch of collinear $S$-balanced configurations and provide an estimate from below on the number of bifurcation instants. In the last part of the paper, by using the continuation method, we explicitly display the bifurcation branches in the case of the three body problem for different choices of the masses.
For planar ($N$+1)-body ($N$,$geq$ 2) problem with a regular $N$-polygon, under the assumption that the ($N$+1)-th body locates at the geometric center of the regular $N$-polygon, we obtain the sufficient and necessary conditions that the $N$+1 bodies can form a central configuration.
We study the relationship between the masses and the geometric properties of central configurations. We prove that in the planar four-body problem, a convex central configuration is symmetric with respect to one diagonal if and only if the masses of the two particles on the other diagonal are equal. If these two masses are unequal, then the less massive one is closer to the former diagonal. Finally, we extend these results to the case of non-planar central configurations of five particles.
Recent developments on three body systems have revealed that dynamics of trajectories passing through collinear configurations can be easily adopted. We analyse the reduction procedure in order to detect the points where collinear configurations are deviating. Then we show that the value of the reduced Hamiltonian can be computed at these points.
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