Related papers: Action minimizing orbits in the n-body problem wit…
We try to prove the existence of choreography solutions for the $n-$body problem on $S^2$. For the three-body problem, we show the existence of the 8-shape orbit on $S^2$.
We prove the existence of a number of smooth periodic motions $u_*$ of the classical Newtonian $N$-body problem which, up to a relabeling of the $N$ particles, are invariant under the rotation group ${\cal R}$ of one of the five Platonic…
We investigate three-body motion in three dimensions under the interaction potential proportional to r^alpha (alpha \neq 0) or log r, where r represents the mutual distance between bodies, with the following conditions: (I) the moment of…
Formulating the equations of motion for cosmological bodies (such as galaxies) in an integral, rather than differential, form has several advantages. Using an integral the mathematical instability at early times is avoided and the boundary…
We first take into account variational problems with periodic boundary conditions, and briefly recall some sufficient conditions for a periodic solution of the Euler-Lagrange equation to be either a directional, a weak, or a strong local…
The simplest solutions of the N-body problem --symmetric relative equilibria-- are shown to be organizing centers from which stem some recently studied classes of periodic solutions. We focus on the relative equilibrium of the equal-mass…
For one 3-body and two 5-body planar choreographies on the same algebraic Lemniscate by Bernoulli we found explicitly a maximal possible set of (particular) Liouville integrals, 7 and 15, respectively, (including the total angular…
In this article, I discuss the motion of $N$ point masses in non-relativistic mechanics, when the interaction between them is purely the Newtonian gravitational interaction, with $N$ greater than or equal to 2. The dynamical equations of…
Consider the Restricted Planar Circular Three Body Problem (RPC3BP), which models the motion of a massless particle (Asteroid) under the gravitational influence of two massive bodies (the primaries) moving on circular orbits. By considering…
We study orbits near collision in a non-autonomous restricted planar four-body problem. This restricted problem consists of a massless particle moving under the gravitational influence due to three bodies following the figure-eight…
We investigate which orbits of an $n$-dimensional torus action on a $2n$-dimensional toric K\"ahler manifold $M$ are minimal. In other words, we study minimal submanifolds appearing as the fibres of the moment map on a toric K\"ahler…
Novel method for semi-analytical solving of equations of a trapped dynamics for a planetoid m4 close to the plane of mutual motion of main bodies around each other (in case of a special type of Bi-Elliptic Restricted 4-Bodies Problem) is…
The restricted (equilateral) four-body problem consists of three bodies of masses m1, m2 and m3 (called primaries) lying in a Lagrangian config- uration of the three-body problem, i,e,. they remain fixed at the apices of an equilateral…
We consider the planar restricted $N$-body problem where the $N-1$ primaries are assumed to be in a central configuration whereas the infinitesimal particle escapes to infinity in a parabolic orbit. We prove the existence of transversal…
We consider a $(1+N)$-body problem in which one particle has mass $m_0 \gg 1$ and the remaining $N$ have unitary mass. We can assume that the body with larger mass (central body) is at rest at the origin, coinciding with the center of mass…
A solution of the n-body problem in R^d is a relative equilibrium if all of the mutual distance between the bodies are constant. In other words, the bodies undergo a rigid motion. Here we investigate the possibility of partially rigid…
In this paper we consider cases of existence of invariant measure, additional first integrals, and Poisson structure in a problem of rigid body's rolling without sliding on plane and sphere. The problem of rigid body's motion on plane was…
Consider a smooth action of $\mathbb R^n$ on a connected manifold $M$, not necessarily compact, of dimension $m$ and rank $k$. Assume that $M$ is not a cylinder. Then there exists an orbit of the action of dimension $<(m+k)/2$. As a…
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…
For some planar Newtonian $N+3$-body problems, we use variational minimization methods to prove the existence of new periodic solutions satisfying that $N$ bodies chase each other on a curve, and the other 3 bodies chase each other on…