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For the Newtonian 4-body problem in space we prove that any zero angular momentum bounded solution suffers infinitely many coplanar instants, that is, times at which all 4 bodies lie in the same plane. This result generalizes a known result…
We consider $(n+1)$ bodies moving under their mutual gravitational attraction in spaces with constant Gaussian curvature $\kappa$. In this system, $n$ primary bodies with equal masses form a relative equilibrium solution with a regular…
The restricted (equilateral) four-body problem consists of three bodies of masses m1, m2 and m3 (called primaries) lying in a Lagrangian configuration of the three-body problem i.e., they remain fixed at the apices of an equilateral…
The three-body problem is reexamined in the framework of general relativity. The Newtonian three-body problem admits Euler's collinear solution, where three bodies move around the common center of mass with the same orbital period and…
The equations of motion of planar elliptic restricted three body problem are transformed to four decoupled Hill's equations. By using the Floquet theorem analytic solution to the oscillator equations with time dependent periodic…
Relativistic systems of particles interacting pairwise at a distance (interactions not mediated by fields) in flat spacetime are studied. It is assumed that the interactions propagate at the speed of light in vacuum and that all masses are…
The equations of motion of two point masses in harmonic coordinates are derived through the third post-Newtonian (3PN) approximation. The problem of self-field regularization (necessary for removing the divergent self-field of point…
The Lagrangian formulation of classical mechanics is widely applicable in solving a vast array of physics problems encountered in the undergraduate and graduate physics curriculum. Unfortunately, many treatments of the topic lack…
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…
A new approach is presented for the problem of optimal impulsive rendezvous of a spacecraft in an inertial frame near a circular orbit in a Newtonian gravitational field. The total characteristic velocity to be minimized is replaced by a…
We consider N-body problems with homogeneous potential $1/r^{2\kappa}$ where $\kappa\in(0,1)$, including the Newtonian case ($\kappa=1/2$). Given $R>0$ and $T>0$, we find a uniform upper bound for the minimal action of paths binding in time…
We consider the $N$-body problem in $\mathbb{R}^d$ with the newtonian potential $1/r$. We prove that for every initial configuration $x_i$ and for every minimizing normalized central configuration $x_0$, there exists a collision-free…
We study a time minimization problem on the group of motions of a plane with admissible control in a half-disk. The considered control system describes a model of a car that can move forward on a plane and turn in place. Optimal…
Using the variational method, Chenciner and Montgomery (2000 Ann. Math. 152 881--901) proved the existence of an eight-shaped periodic solution of the planar three-body problem with equal masses. Just after the discovery, Gerver have…
In this paper, we apply the variational method with the Structural Prescribed Boundary Conditions (SPBC) to prove the existence of periodic and quasi-periodic solutions for planar four-body problem with $m_1=m_3$ and $m_2=m_4$. A path…
In this work, we study the motions in the region around the equilateral Lagrangian equilibrium points L4 and L5, in the framework of the Circular Planar Restricted Three-Body Problem (hereafter, CPRTBP). We design a semi-analytic approach…
Equations are derived for the shape of a hypersurface in $\mathbb{R}^N$ for which a rigid motion yields a minimal surface in $\mathbb{R}^{N+1}$. Some elementary, but unconventional, aspects of the classical case $N=2$ (solved by H.F. Scherk…
In the spatial circular restricted three-body problem librational motion around the Lagrangian points $L_4$ and $L_5$ exists up to high inclinations of the massless object. We report the results of numerical investigations on the stability…
We take into account the Coulomb (N + 1)-body problem with N = 12, 24, 60. One of the particles has positive charge Q > 0, and the remaining N have all the same negative charge q < 0. These particles move under the Coulomb force, and the…
A solvable many-body problem in the plane is exhibited. It is characterized by rotation-invariant Newtonian (``acceleration equal force'') equations of motion, featuring one-body (``external'') and pair (``interparticle'') forces. The…