Related papers: The only syzygy-free solution is Lagrange's

Consider the planar three-body problem with masses positive $m_1,m_2,m_3$ position vector $q(t) = (q_1(t),q_2(t),q_3(t))\in\mathbb{R}^6$. Let $$U(q) = \frac{m_1m_2}{r_{12}}+\frac{m_1m_3}{r_{13}}+\frac{m_2m_3}{r_{23}}$$ where…

We consider the $n$--body problem defined on surfaces of constant negative curvature. For the case of $n$--equal masses we prove that the hyperbolic relative equilibria with a regular polygonal shape do not exist. In particular the…

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…

The closed form solution is found for the fully nonlinear dynamics of the gyroscope with a fixed point at the tip. The solution is found by using Cardano's formulae to factor a cubic, in the case where all roots are known to be real. From…

The relative equilibria for the spherical, finite density 3 body problem are identified. Specifically, there are 28 distinct relative equilibria in this problem which include the classical 5 relative equilibria for the point-mass 3-body…

We introduce a circular restricted charged three-body problem on the plane. In this model, the gravitational and Coulomb forces, due to the primary bodies, act on a test particle; the net force exerted by some primary body on the test…

We study the problem of motion of a relativistic, ideal elastic solid with free surface boundary by casting the equations in material form ("Lagrangian coordinates"). By applying a basic theorem due to Koch, we prove short-time existence…

Novel classes of dynamical systems are introduced, including many-body problems characterized by nonlinear equations of motion of Newtonian type ("acceleration equals forces") which determine the motion of points in the complex plane. These…

This paper presents a study of the isosceles problem resulting by a perturbation of Euler's collinear solution under Newtonian gravitational attraction of three bodies in space. After the Hamiltonian was obtained, a circumference of…

When we use variational methods to study the Newtonian $N$-body problem, the main problem is how to avoid collisions. C.Marchal got a remarkable result, that is, a path minimizing the Lagrangian action functional between two given…

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…

For $n$-body problem with arbitrary positive masses, we prove there are regularizable collinear periodic solutions for any ordering of the masses, going from a simultaneous binary collision to another in half of a period with half of the…

Starting from a relativistic s-wave scattering length model for the two particle input we construct an unambiguous, unitary solution of the relativistic three body problem given only the masses $m_a,m_b,m_c$ and the masses of the two body…

The existence of stationary solutions to the Einstein-Vlasov system which are axially symmetric and have non-zero total angular momentum is shown. This provides mathematical models for rotating, general relativistic and asymptotically flat…

Solutions to gravity with quadratic Lagrangians are found for the simple case where the only nonconstant metric component is the lapse $N$ and the Riemann tensor takes the form $R^{t}_{.itj}=-k_{i}k_{j}, i,j=1,2,3$; thus these solutions…

The exact solution to the Schr\"{o}dinger equation for the rigid body with the given angular momentum and parity is obtained. Since the quantum rigid body can be thought of as the simplest quantum three-body problem where the internal…

We give a complete symbolic dynamics description of the dynamics of Euler's problem of two fixed centers. By analogy with the 3-body problem we use the collinearities (or syzygies) of the three bodies as symbols. We show that motion without…

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…

In this paper, an approach is developed to solve the three body problem involving masses which posses spherical symmetry. The problem dates back to the times of Poincare, and is undoubtedly one of the oldest of unsolved problems of…

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…