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Manuele Santoprete

Publications and source records attributed to Manuele Santoprete.

At least 37 records · Page 2Linked to original sources
arXiv2012

Relative Equilibria in the Four-Vortex Problem with Two Pairs of Equal Vorticities

We examine in detail the relative equilibria in the four-vortex problem where two pairs of vortices have equal strength, that is, Γ_1 = Γ_2 = 1 and Γ_3 = Γ_4 = m where m is a nonzero real parameter. One main result is that for m > 0, the convex configurations all contain a line of symmetry, forming a rhombus or an isosceles trapezoid. The rhombus solutions exist for all m but the isosceles trapezoid case exists only when m is positive. In fact, there exist asymmetric convex configurations when m < 0. In contrast to the Newtonian four-body problem with two equal pairs of masses, where the symmetry of all convex central configurations is unproven, the equations in the vortex case are easier to handle, allowing for a complete classification of all solutions. Precise counts on the number and type of solutions (equivalence classes) for different values of m, as well as a description of some of the bifurcations that occur, are provided. Our techniques involve a combination of analysis and modern and computational algebraic geometry.

math.CA↗
arXiv2012

Regularization of the Kepler problem on the Sphere

In this paper we regularize the Kepler problem on $S^3$ in several different ways. First, we perform a Moser-type regularization. Then, we adapt the Ligon-Schaaf regularization to our problem. Finally, we show that the Moser regularization and the Ligon-Schaaf map we obtained can be understood as the composition of the corresponding maps for the Kepler problem in Euclidean space and the gnomonic transformation.

math.DS↗
arXiv2009

Smoothed dynamics in the central field problem

Consider the motion of a material point of unit mass in a central field determined by a homogeneous potential of the form $(-1/r^α)$, $α>0,$ where $r$ being the distance to the centre of the field. Due to the singularity at $r=0,$ in computer-based simulations, usually, the potential is replaced by a similar potential that is smooth, or at least continuous. In this paper, we compare the global flows given by the smoothed and non-smoothed potentials. It is shown that the two flows are topologically equivalent for $α< 2,$ while for $α\geq 2,$ smoothing introduces fake orbits. Further, we argue that for $α\geq 2,$ smoothing should be applied to the amended potential $c/(2r^2)-1/r^α,$ where $c$ denotes the angular momentum constant.

math-ph↗
arXiv2009

Central Configurations and Total Collisions for Quasihomogeneous n-Body Problems

We consider $n$-body problems given by potentials of the form ${α\over r^a}+{β\over r^b}$ with $a,b,α,β$ constants, $0\le a<b$. To analyze the dynamics of the problem, we first prove some properties related to central configurations, including a generalization of Moulton's theorem. Then we obtain several qualitative properties for collision and near-collision orbits in the Manev-type case $a=1$. At the end we point out some new relationships between central configurations, relative equilibria, and homothetic solutions.

math-ph↗
arXiv2009

Saari's Homographic Conjecture of the Three-Body Problem

Saari's homographic conjecture, which extends a classical statement proposed by Donald Saari in 1970, claims that solutions of the Newtonian $n$-body problem with constant configurational measure are homographic. In other words, if the mutual distances satisfy a certain relationship, the configuration of the particle system may change size and position but not shape. We prove this conjecture for large sets of initial conditions in three-body problems given by homogeneous potentials, including the Newtonian one. Some of our results are true for $n\ge 3$.

math-ph↗
arXiv2009

Linear stability of the Lagrangian triangle solutions for quasihomogeneous potentials

In this paper we study the linear stability of the relative equilibria for homogeneous and quasihomogeneous potentials. Firstly, in the case the potential is a homogeneous function of degree $-a$, we find that any relative equilibrium of the $n$-body problem with $a>2$ is spectrally unstable. We also find a similar condition in the quasihomogeneous case. Then we consider the case of three bodies and we study the stability of the equilateral triangle relative equilibria. In the case of homogeneous potentials we recover the classical result obtained by Routh in a simpler way. In the case of quasihomogeneous potentials we find a generalization of Routh inequality and we show that, for certain values of the masses, the stability of the relative equilibria depends on the size of the configuration.

math-ph↗
arXiv2009

A Counterexample to a Generalized Saari's Conjecture with a Continuum of Central Configurations

In this paper we show that in the $n$-body problem with harmonic potential one can find a continuum of central configurations for $n=3$. Moreover we show a counterexample to an interpretation of Jerry Marsden Generalized Saari's conjecture. This will help to refine our understanding and formulation of the Generalized Saari's conjecture, and in turn it might provide insight in how to solve the classical Saari's conjecture for $n\geq 4$.

math-ph↗
arXiv2009

The Kepler Problem with Anisotropic Perturbations

We study a 2-body problem given by the sum of the Newtonian potential and an anisotropic perturbation that is a homogeneous function of degree $-β$, $β\ge 2$. For $β>2$, the sets of initial conditions leading to collisions/ejections and the one leading to escapes/captures have positive measure. For $β>2$ and $β\ne 3$, the flow on the zero-energy manifold is chaotic. For $β=2$, a case we prove integrable, the infinity manifold of the zero-energy level is a disconnected set, which has heteroclinic connections with the collision manifold.

math-ph↗
arXiv2009

Saari's Conjecture for the Collinear $n$-Body Problem

In 1970 Don Saari conjectured that the only solutions of the Newtonian $n$-body problem that have constant moment of inertia are the relative equilibria. We prove this conjecture in the collinear case for any potential that involves only the mutual distances. Furthermore, in the case of homogeneous potentials, we show that the only collinear and non-zero angular momentum solutions are homographic motions with central configurations.

math-ph↗
arXiv2009

Convex Four Body Central Configurations with Some Equal Masses

We prove that there is a unique convex non-collinear central configuration of the planar Newtonian four-body problem when two equal masses are located at opposite vertices of a quadrilateral and, at most, only one of the remaining masses is larger than the equal masses. Such central configuration posses a symmetry line and it is a kite shaped quadrilateral. We also show that there is exactly one convex non-collinear central configuration when the opposite masses are equal. Such central configuration also posses a symmetry line and it is a rhombus.

math-ph↗
arXiv2009

Rosette Central Configurations, Degenerate central configurations and bifurcations

In this paper we find a class of new degenerate central configurations and bifurcations in the Newtonian $n$-body problem. In particular we analyze the Rosette central configurations, namely a coplanar configuration where $n$ particles of mass $m_1$ lie at the vertices of a regular $n$-gon, $n$ particles of mass $m_2$ lie at the vertices of another $n$-gon concentric with the first, but rotated of an angle $π/n$, and an additional particle of mass $m_0$ lies at the center of mass of the system. This system admits two mass parameters $μ=m_0/m_1$ and $\ep=m_2/m_1$. We show that, as $μ$ varies, if $n> 3$, there is a degenerate central configuration and a bifurcation for every $\ep>0$, while if $n=3$ there is a bifurcations only for some values of $ε$.

math-ph↗
arXiv2009

Central Configurations of the Five-Body Problem with Equal Masses

In this paper we present a complete classification of the isolated central configurations of the five-body problem with equal masses. This is accomplished by using the polyhedral homotopy method to approximate all the isolated solutions of the Albouy-Chenciner equations. The existence of exact solutions, in a neighborhood of the approximated ones, is then verified using the Krawczyk method. Although the Albouy-Chenciner equations for the five-body problem are huge, it is possible to solve them in a reasonable amount of time.

math-ph↗
arXiv2009

Block regularization of the Kepler problem on surfaces of revolution with positive constant curvature

We consider the Kepler problem on surfaces of revolution that are homeomorphic to $S^2$ and have constant Gaussian curvature. We show that the system is maximally superintegrable, finding constants of motion that generalize the Runge-Lentz vector. Then, using such first integrals, we determine the class of surfaces that lead to block-regularizable collision singularities. In particular we show that the singularities are always regularizable if the surfaces are spherical orbifolds of revolution with constant curvature.

math-ph↗
arXiv2008

The n-body problem in spaces of constant curvature

We generalize the Newtonian n-body problem to spaces of curvature k=constant, and study the motion in the 2-dimensional case. For k>0, the equations of motion encounter non-collision singularities, which occur when two bodies are antipodal. This phenomenon leads, on one hand, to hybrid solution singularities for as few as 3 bodies, whose corresponding orbits end up in a collision-antipodal configuration in finite time; on the other hand, it produces non-singularity collisions, characterized by finite velocities and forces at the collision instant. We also point out the existence of several classes of relative equilibria, including the hyperbolic rotations for k<0. In the end, we prove Saari's conjecture when the bodies are on a geodesic that rotates elliptically or hyperbolically. We also emphasize that fixed points are specific to the case k>0, hyperbolic relative equilibria to k<0, and Lagrangian orbits of arbitrary masses to k=0--results that provide new criteria towards understanding the large-scale geometry of the physical space.

math.DS↗
arXiv2002

Symmetric Periodic Solutions of the Anisotropic Manev Problem

We consider the Manev Potential in an anisotropic space, i.e., such that the force acts differently in each direction. Using a generalization of the Poincare' continuation method we study the existence of periodic solutions for weak anisotropy. In particular we find that the symmetric periodic orbits of the Manev system are perturbed to periodic orbits in the anisotropic problem.

nlin.CD↗