Related papers: Torus knot choreographies in the $n$-body problem

General properties of the three-body problem in a model of modified dynamics are investigated. It is shown that the three-body problem in this model shares some characters with the similar problem in Newtonian dynamics. Moreover, the planar…

Periodic solutions of the planar $N$-body problem determine braids through the trajectory of $N$ bodies. Braid types can be used to classify periodic solutions. According to the Nielsen-Thurston classification of surface automorphisms,…

Analytical methods are used to prove the existence of a periodic, symmetric solution with singularities in the planar 4-body problem. A numerical calculation and simulation are used to generate the orbit. The analytical method easily…

We explore the $n$-body problem, $n\geq 3,$ on a surface of revolution with a general interaction depending on the pairwise geodesic distance. Using the geometric methods of classical mechanics we determine a large set of properties. In…

In this work we have studied the singular behaviour of gravitational theories with non symmetric connections. For this purpose we introduce a new criteria for the appearance of singularities based on the existence of black/white hole…

We show that for many classical knots one can find generalized torsion in the fundamental group of its complement, commonly called the knot group. It follows that such a group is not bi-orderable. Examples include all torus knots, the…

We consider two natural Lagrangian intersection problems in the context of symplectic toric manifolds: displaceability of torus orbits and of a torus orbit with the real part of the toric manifold. Our remarks address the fact that one can…

We consider an effective action of a compact (n-1)-torus on a smooth 2n-manifold with isolated fixed points. We prove that under certain conditions the orbit space is a closed topological manifold. In particular, this holds for certain…

We provide in this paper the discrete equations of motion for the newtonian $n$-body problem deduced from the quantum calculus of variations (Q.C.V.) developed in \cite{Cre,CFT,RS1,RS2}. These equations are brought into the usual lagrangian…

The interaction between spin and gravitational waves causes spinning bodies to deviate from their geodesics. In this work, we obtain the analytic solution of the Mathisson--Papapetrou--Dixon equations at linear order in the spin for plane…

This study presents a general alternative scheme of the procedure and necessary conditions for solving the $n$-body problem. The presented solution is not a solution of the classical problem, where the initial conditions of positions and…

Knot concordance plays a crucial role in the low dimensional topology. We propose a very elementary techniques which allows one to construct a lot of sliceness obstructions for knots in the full torus. Our approach deals with group…

An algorithm is presented for numerical computation of choreographies in the plane in a Newtonian potential and on the sphere in a cotangent potential. It is based on stereographic projection, approximation by trigonometric polynomials, and…

We consider the $N$-body problem of celestial mechanics in spaces of nonzero constant curvature. Using the concept of locked inertia tensor, we compute the moment of inertia for systems moving on spheres and hyperbolic spheres and show that…

We study singularities of the n-body problem in spaces of constant curvature and generalize certain results due to Painleve, Weierstrass, and Sundman. For positive curvature, some of our proofs use the correspondence between total collision…

The internal space for a molecule, atom, or other n-body system can be conveniently parameterised by 3n-9 kinematic angles and three kinematic invariants. For a fixed set of kinematic invariants, the kinematic angles parameterise a…

We prove the existence of at least two geometrically different periodic solution with winding number N for the forced relativistic pendulum. The instability of a solution is also proved. The proof is topological and based on the version of…

The time-dependent restricted $(n+1)$-body problem concerns the study of a massless body (satellite) under the influence of the gravitational field generated by $n$ primary bodies following a periodic solution of the $n$-body problem. We…

In a remarkable paper of 2003 by Fujiwara et al. \cite{Fujiwara2003}, a figure-eight three-body choreography on the algebraic lemniscate by Bernoulli was discovered. Such a choreography was found to be driven by the action of a pairwise…

Bodies with the nonspherical tensor of inertia exhibit a variety of rotational motion patterns, including chaotic motion, stable periodic (quasi-periodic) rotation, unstable rotation around the direction close to the body's second principal…