The 3-body problem in $\mathbb{R}^4$ has 24 dimensions and is invariant under translations and rotations. We do the full symplectic symmetry reduction and obtain a reduced Hamiltonian in local symplectic coordinates on a reduced phase space with 8 dimensions. The Hamiltonian depends on two parameters $μ_1 > μ_2 \ge 0$, related to the conserved angular momentum. The limit $μ_2 \to 0$ corresponds to the 3-dimensional limit. We show that the reduced Hamiltonian has relative equilibria that are local minima and hence Lyapunov stable when $μ_2$ is sufficiently small. This proves the existence of balls of initial conditions of full dimension that do not contain any orbits that are unbounded.
Using geometric mechanics methods, we examine aspects of the dynamics of n mass points in $\mathbb{R}^4$ with a general pairwise potential. We investigate the central force problem, set up the n-body problem and discuss certain properties of relative equilibria. We describe regular n-gons in $\mathbb{R}^4$ and when the masses are equal, we determine the invariant manifold of motions with regular n-gon configurations. In the case n=3 we reduce the dynamics to a six degrees of freedom system and we show that for generic potentials and momenta, relative equilibria with equilateral configuration a…
We prove that the fixed points of the curved 3-body problem and their associated relative equilibria are Lyapunov stable if the solutions are restricted to $\mathbb S^1$, but unstable if the bodies are considered in $\mathbb S^2$.
We construct the hyperbolic plane with its geodesic flow as the scale plus symmetry reduction of a three-body problem in the Euclidean plane. The potential is $-I/Δ^2$ where $I$ is the triangle's moment of inertia and $Δ$ its area. The reduction method uses the Jacobi-Maupertuis metric, following the author's earlier paper "Putting Hyperbolic Pants on a Three-body Problem".
The classical equations of the Newtonian 3-body problem do not only define the familiar 3-dimensional motions. The dimension of the motion may also be 4, and cannot be higher. We prove that in dimension 4, for three arbitrary positive masses, and for an arbitrary value (of rank 4) of the angular momentum, the energy possesses a minimum, which corresponds to a motion of relative equilibrium which is Lyapunov stable when considered as an equilibrium of the reduced problem. The nearby motions are nonsingular and bounded for all time. We also describe the full family of relative equilibria, and sh…
In his fondamental "Essay on the 3-body problem", Lagrange, well before Jacobi's "reduction of the node", carries out the first complete reduction of symetries. Discovering the so-called homographic motions, he shows that they necessarily take place in a fixed plane. The true nature of this reduction is revealed if one considers the n-body problem in an euclidean space of arbitrary dimension. The actual dimension of the ambiant space then appears as a constraint, namely the angular momentum bivector's degeneracy. The main part of this survey is a detailed description of the results obtained in…
We carry out a sequence of coordinate changes for the planar three-body problem which successively eliminate the translation and rotation symmetries, regularize all three double collision singularities and blow-up the triple collision. Parametrizing the configurations by the three relative position vectors maintains the symmetry among the masses and simplifies the regularization of binary collisions. Using size and shape coordinates facilitates the reduction by rotations and the blow-up of triple collision while emphasizing the role of the shape sphere. By using homogeneous coordinates to desc…
In this paper, we consider the elliptic collinear solutions of the classical $n$-body problem, where the $n$ bodies always stay on a straight line, and each of them moves on its own elliptic orbit with the same eccentricity. Such a motion is called an elliptic Euler-Moulton collinear solution. Here we prove that the corresponding linearized Hamiltonian system at such an elliptic Euler-Moulton collinear solution of $n$-bodies splits into $(n-1)$ independent linear Hamiltonian systems, the first one is the linearized Hamiltonian system of the Kepler $2$-body problem at Kepler elliptic orbit, and…
It is found explicitly 5 Liouville integrals in addition to total angular momentum which Poisson commute with Hamiltonian of 3-body Newtonian Gravity in ${\mathbb R}^3$ along the Remarkable Figure-8-shape trajectory discovered by Moore-Chenciner-Montgomery. It is shown they become constants of motion along this trajectory. Hence, 3-body choreographic motion on Figure-8-shape trajectory in ${\mathbb R}^3$ Newtonian gravity (Moore, 1993), as well as in ${\mathbb R}^2$ modified Newtonian gravity by Fujiwara et al, 2003, is maximally superintegrable. It is conjectured that any 3-body potential the…
A rational map with good reduction in the field $\mathbb{Q}\_p$ of $p$-adic numbers defines a $1$-Lipschitz dynamical system on the projective line $\mathbb{P}^1(\mathbb{Q}\_p)$ over $\mathbb{Q}\_p$. The dynamical structure of such a system is completely described by a minimal decomposition. That is to say, $\mathbb{P}^1(\mathbb{Q}\_p)$ is decomposed into three parts: finitely many periodic orbits; finite or countably many minimal subsystems each consisting of a finite union of balls; and the attracting basins of periodic orbits and minimal subsystems. For any prime $p$, a criterion of minimal…
The aim of the present work is to reduce the secular solution around the triangular equilibrium points to periodic solution in the frame work of the generalized restricted thee-body problem. This model is generalized in sense that both the primaries are oblate and radiating as well as the gravitational potential from a belt. We show that the linearized equation of motion of the infinitesimal body around the triangular equilibrium points has a secular solution when the value of mass ratio equals the critical mass value. Moreover, we reduce this solution to periodic solution, as well as some num…
Gauss' classical reduction theory for indefinite binary quadratic forms over $\mathbb{Z}$ has originally been proven by means of purely algebraic and arithmetic considerations. It was later discovered that this reduction theory is closely related to a certain symbolic dynamics for the geodesic flow on the modular surface, and hence can also be deduced geometrically. In this article, we use certain symbolic dynamics for the geodesic flow on Hecke triangle surfaces (also the non-arithmetic ones) to develop reduction theories for the indefinite binary quadratic forms associated to Hecke triangle …
We denote by $\mathbb{M}^2_R$ a two dimensional space of constant positive Gaussian curvature. With methods of Möbius geometry and using the classification of the Möbius group of automorphisms ${\rm \bf Mob}_2 \, (\hat{\mathbb{C}})$ of the Riemman sphere $\hat{\mathbb{C}}=\mathbb{M}_R^2 \cup \{\infty\}$, we give algebraic conditions for the existence of Möbius solutions on $\hat{\mathbb{C}}$, getting a complete classification of them. We show several families of this kind of solutions.
We consider the dynamics and symplectic reduction of the 2-body problem on a sphere of arbitrary dimension. It suffices to consider the case for when the sphere is 3-dimensional and where we take the group of symmetries to be $SO(4)$. As the 3-sphere is a group, both left and right multiplication on itself are commuting symmetries which together generate the full symmetry group. This gives rise to a notion of left and right momenta for the problem, and allows for a reduction in stages, first by the left and then the right, or vice versa. The intermediate reduced spaces obtained by left or righ…
We consider the 3-dimensional gravitational $n$-body problem, $n\ge 2$, in spaces of constant Gaussian curvature $κ\ne 0$, i.e.\ on spheres ${\mathbb S}_κ^3$, for $κ>0$, and on hyperbolic manifolds ${\mathbb H}_κ^3$, for $κ<0$. Our goal is to define and study relative equilibria, which are orbits whose mutual distances remain constant in time. We also briefly discuss the issue of singularities in order to avoid impossible configurations. We derive the equations of motion and define six classes of relative equilibria, which follow naturally from the geometric properties of ${\mathbb S}_κ^3$ and…
We restate the semistable reduction theorem from geometric invariant theory in the context of spaces of morphisms on $\mathbb{P}^{n}$. For every complete curve $C$ downstairs, we get a $\mathbb{P}^{n}$-bundle on an abstract curve $D$ mapping finite-to-one onto $C$, whose trivializations correspond to not necessarily complete curves upstairs with morphisms corresponding to identifying each fiber with the morphism the point represents. Finding a trivial bundle is equivalent to finding a complete $D$ upstairs mapping finite-to-one onto $C$; we prove that in every space of morphisms, there exists …
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…
In this document the author proves that several problems in data-driven numerical approximation of dynamical systems in $\mathbb{C}^n$, can be reduced to the computation of a family of constrained matrix representations of elements of the group algebra $\mathbb{C}[\mathbb{Z}/m]$ in $\mathbb{C}^{n\times n}$, factoring through the commutative algebra $Circ(m)$ of circulant matrices in $\mathbb{C}^{m\times m}$, for some integers $m\leq n$. The solvability of the previously described matrix representation problems is studied. Some connections of the aforementioned results, with numerical analysis …
Let $K$ be a number field, let $S$ be a finite set of places of $K$, and let $R_S$ be the ring of $S$-integers of $K$. A $K$-morphism $f:\mathbb{P}^1_K\to\mathbb{P}^1_K$ has simple good reduction outside $S$ if it extends to an $R_S$-morphism $\mathbb{P}^1_{R_S}\to\mathbb{P}^1_{R_S}$. A finite Galois invariant subset $X\subset\mathbb{P}^1_K(\bar{K})$ has good reduction outside $S$ if its closure in $\mathbb{P}^1_{R_S}$ is étale over $R_S$. We study triples $(f,Y,X)$ with $X=Y\cup f(Y)$. We prove that for a fixed $K$, $S$, and $d$, there are only finitely many $\text{PGL}_2(R_S)$-equivalence cl…
We consider a perturbation of a central force problem of the form \begin{equation*} \ddot x = V'(|x|) \frac{x}{|x|} + \varepsilon \,\nabla_x U(t,x), \quad x \in \mathbb{R}^{2} \setminus \{0\}, \end{equation*} where $\varepsilon \in \mathbb{R}$ is a small parameter, $V\colon (0,+\infty) \to \mathbb{R}$ and $U\colon \mathbb{R} \times (\mathbb{R}^{2} \setminus \{0\}) \to \mathbb{R}$ are smooth functions, and $U$ is $τ$-periodic in the first variable. Based on the introduction of suitable time-maps (the radial period and the apsidal angle) for the unperturbed problem ($\varepsilon=0$) and of an as…
Based on a concept link between the partial reduction procedure of the reduction of the rotational symmetry of the N-body problem with the symplectic cross-section theorem of Guillemin-Sternberg, we present alternative proofs of the symplecticity of Delaunay and Deprit coordinates in celestial mechanics.
Let $K$ be a number field and $v$ a non archimedean valuation on $K$. We say that an endomorphism $Φ\colon \mathbb{P}_1\to \mathbb{P}_1$ has good reduction at $v$ if there exists a model $Ψ$ for $Φ$ such that $\degΨ_v$, the degree of the reduction of $Ψ$ modulo $v$, equals $\degΨ$ and $Ψ_v$ is separable. We prove a criterion for good reduction that is the natural generalization of a result due to Zannier in \cite{Uz3}. Our result is in connection with other two notions of good reduction, the simple and the critically good reduction. The last part of our article is dedicated to prove a characte…
We revisit polygonal positive elliptic rotopulsator solutions and polygonal negative elliptic rotopulsator solutions of the $n$-body problem in $\mathbb{H}^{3}$ and $\mathbb{S}^{3}$ and prove existence of these solutions, prove that the masses of these rotopulsators have to be equal if the rotopulsators are of nonconstant size and show that the number of negative elliptic relative equilibria of this type is finite, as is the number of positive elliptic relative equilibria if an upper bound on the size of the relative equilibrium is imposed. Additionally, we prove that a class of negative hyper…
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.
