Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userBifurcations of balanced configurations for the Newtonian $n$-body problem in $mathbb R^4$
115
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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 Newtonian (gravitational) $n$-body problem in the Euclidean $d$-dimensional space, $dge 2$, the simplest possible periodic solutions are provided by circular relative equilibria, (RE) for short, namely solutions in which each body rigidly rotates about the center of mass and the configuration of the whole system is constant in time and central (or, more generally, balanced) configuration. For $dle 3$, the only possible (RE) are planar, but in dimension four it is possible to get truly four dimensional (RE). A classical problem in celestial mechanics aims at relating the (in-)stability properties of a (RE) to the index properties of the central (or, more generally, balanced) configuration generating it. In this paper, we provide sufficient conditions that imply the spectral instability of planar and non-planar (RE) in $mathbb R^4$ generated by a central configuration, thus answering some of the questions raised in cite[Page 63]{Moe14}. As a corollary, we retrieve a classical result of Hu and Sun cite{HS09} on the linear instability of planar (RE) whose generating central configuration is non-degenerate and has odd Morse index, and fix a gap in the statement of cite[Theorem 1]{BJP14} about the spectral instability of planar (RE) whose (possibly degenerate) generating central configuration has odd Morse index. The key ingredients are a new formula of independent interest that allows to compute the spectral flow of a path of symmetric matrices having degenerate starting point, and a symplectic decomposition of the phase space of the linearized Hamiltonian system along a given (RE) which is inspired by Meyer and Schmidts planar decomposition cite{MS05} and which allows us to rule out the uninteresting part of the dynamics corresponding to the translational and (partially) to the rotational symmetry of the problem.
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.
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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
