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Euler's three-body problem is the problem of solving for the motion of a particle moving in a Newtonian potential generated by two point sources fixed in space. This system is integrable in the Liouville sense. We consider the Euler problem…
In this article, we identify the 5-dimensional analogue of the finite energy foliations introduced by Hofer--Wysocki--Zehnder for the study of 3-dimensional Reeb flows, and show that these exist for the spatial circular restricted…
We present numerical simulations of the gravitational three-body problem, in which three particles lie at rest close to the vertices of an equilateral triangle. In the unperturbed problem, the three particles fall towards the center of mass…
We consider a problem of mass points interacting gravitationally whose motion is subjected to certain holonomic constraints. The motion of points is restricted to certain curves and surfaces. We illustrate the complicated behaviour of…
Errors in numerical simulations of gravitating systems can be magnified exponentially over short periods of time. Numerical shadowing provides a way of demonstrating that the dynamics represented by numerical simulations are representative…
We study the dynamics of two bodies moving on elliptic Keplerian orbits around a fixed center of attraction and interacting only by means of elastic or inelastic collisions. We show that there exists a bounded invariant region: for suitable…
We study the long-time behavior of an elliptic rigid body which is allowed to vertically translate and rotate in a 2D unbounded channel under the action of a Poiseuille flow at large distances. The motion of the fluid is modelled by the…
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…
We study the stationary points of the hierarchical three body problem in the planetary limit (m_2, m_3 << m_1) at both the quadrupole and octupole orders. We demonstrate that the extension to octupole order preserves the principal…
We propose a contact-topological approach to the spatial circular restricted three-body problem, for energies below and slightly above the first critical energy value. We prove the existence of a circle family of global hypersurfaces of…
We develop a framework based on energy kicks for the evolution of high-eccentricity long-period orbits with Jacobi constant close to 3 in the restricted circular planar three-body problem where the secondary and primary masses have mass…
The sum of elliptic integrals simultaneously determines orbits in thr Kepler problem and the addition of divisors on elliptic curves. Periodic motion of a body in physical space is defined by symmetries, whereas periodic motion of divisors…
The 2:1 mean motion resonance orbit was integrated at the restricted planar 3-body problem in absolute frame. Orbit of Jupiter was assumed circular. Initial Jupiter longitude was assumed zero. The Runge-Kutta method was used. The start of…
We consider the restricted n + 1-body problem of Newtonian mechanics. For periodic, planar configurations of n bodies which is symmetric under rotation by a fixed angle, the z-axis is invariant. We consider the effect of placing a massless…
We investigate the motion of a massless body interacting with the Maxwell relative equilibrium, which consists of n bodies of equal mass at the vertices of a regular polygon that rotates around a central mass. The massless body has three…
In this work, we revisit the planar restricted four-body problem to study the dynamics of an infinitesimal mass under the gravitational force produced by three heavy bodies with unequal masses, forming an equilateral triangle configuration.…
The results of an extensive numerical study of the periodic orbits of planar, elliptic restricted three-body planetary systems consisting of a star, an inner massive planet and an outer mass-less body in the external 1:2 mean-motion…
In this paper we review a recently developed approximate method for investigation of dynamics of compressible ellipsoidal figures. Collapse and subsequent behaviour are described by a system of ordinary differential equations for time…
We first derive the Rayleigh-Schr\"odinger many-body perturbation theory up to third order (RSPT3) for Hamiltonians with three-body interaction. The structure of closed-shell nuclei in a wide mass range from 4He to 48Ca has been…
The dynamics near the Lagrange equilibria $L_1$ and $L_2$ of the Circular Restricted Three-body Problem has gained attention in the last decades due to its relevance in some topics such as the temporary captures of comets and asteroids and…