Florin Diacu

University of Victoria | UVIC · Department of Mathematics and Statistics

We study the variational property of the periodic Kepler orbits on the sphere, the plane and the hyperbolic plane. We first classify the orbits by the two constants of motion: the energy and the angular momentum. Then, we characterize the local variational property of the closed orbits by computing the Maslov-type indices. Finally, we study the glo...

We consider the $N$-body problem of celestial mechanics in spaces of nonzero constant curvature. Using the concept of locked inertia tensor, we compute the moment of inertia for systems moving on spheres and hyperbolic spheres and show that we can recover the classical definition in the Euclidean case. After proving some criteria for the existence...

We consider the curved 4-body problems on spheres and hyperbolic spheres. After obtaining a criterion for the existence of quadrilateral configurations on the equator of the sphere, we study two restricted 4-body problems, one in which two masses are negligible and another in which only one mass is negligible. In the former, we prove the evidence s...

We generalize the curved $N$-body problem to spheres and hyperbolic spheres whose curvature $\kappa$ varies in time. Unlike in the particular case when the curvature is constant, the equations of motion are non-autonomous. We first briefly consider the analogue of the Kepler problem and then investigate the homographic orbits for any number of bodi...

We obtain a natural extension of the Vlasov-Poisson system for stellar dynamics to spaces of constant Gaussian curvature $\kappa\ne 0$: the unit sphere $\mathbb S^2$, for $\kappa>0$, and the unit hyperbolic sphere $\mathbb H^2$, for $\kappa<0$. These equations can be easily generalized to higher dimensions. When the particles move on a geodesic, th...

We prove that the fixed points of the curved 3-body problem and their associated relative equilibria are Lyapunov stable if the solutions are restricted to S1, but their linear stability depends on the angular velocity if the bodies are considered on S2. More precisely, the associated relative equilibria are linearly stable if and only if the angul...

We prove that the fixed points of the curved 3-body problem and their associated relative equilibria are Lyapunov stable if the solutions are restricted to $\mathbb S^1$, but unstable if the bodies are considered in $\mathbb S^2$.

We ask whether Hamiltonian vector fields defined on spaces of constant Gaussian curvature $\kappa$ (spheres, for $\kappa>0$, and hyperbolic spheres, for $\kappa<0$), pass continuously through the value $\kappa=0$ if the potential functions $U_\kappa, \kappa\in\mathbb R$, that define them satisfy the property $\lim_{\kappa\to 0}U_\kappa=U_0$, where...

We consider the 3-body problem of celestial mechanics in Euclidean, elliptic, and hyperbolic spaces, and study how the Lagrangian (equilateral) relative equilibria bifurcate when the Gaussian curvature varies. We thus prove the existence of new classes of orbits. In particular, we find some families of isosceles triangles, which occur in elliptic s...

The idea that geometry and physics are intimately related made its way in human thought during the early part of the nineteenth century. Gauss measured the angles of a triangle formed by three mountain peaks near Göttingen, Germany, apparently hoping to learn whether the universe has positive or negative curvature, but the inevitable observational...

The idea that geometry and physics are intimately related made its way in human thought during the early part of the nineteenth century. Gauss measured the angles of a triangle formed by three mountain peaks near Göttingen, Germany, apparently hoping to learn whether the universe has positive or negative curvature, but the inevitable observational...

We consider the 3-body problem in 3-dimensional spaces of nonzero constant Gaussian curvature and study the relationship between the masses of the Lagrangian relative equilibria, which are orbits that form a rigidly rotating equilateral triangle at all times. There are three classes of Lagrangian relative equilibria in 3-dimensional spaces of const...

We provide the differential equations that generalize the classical N-body problem of celestial mechanics to spaces of constant Gaussian curvature, k, for all k real. In previous studies, the equations of motion made sense only for k nonzero. The system derived here does more than just include the Euclidean case in the limit when k tends to 0; the...

We consider the 4-body problem in spaces of constant curvature and study the existence of spherical and hyperbolic rectangular solutions, i.e. equiangular quadrilateral motions on spheres and hyperbolic spheres. We focus on relative equilibria (orbits that maintain constant mutual distances) and rotopulsators (configurations that rotate and change...

We consider the N-body problem in spaces of constant curvature and study its rotopulsators, i.e.\ solutions for which the configuration of the bodies rotates and changes size during the motion. Rotopulsators fall naturally into five groups: positive elliptic, positive elliptic-elliptic, negative elliptic, negative hyperbolic, and negative elliptic-...

In this chapter we will introduce some concepts needed for the derivation of the equations of motion of the N-body problem in spaces of constant curvature as well as for the study of the relative equilibria, which are special classes of orbits that we will investigate later. We will start by introducing a model for hyperbolic geometry, usually attr...

The main purpose of this chapter is to derive the equations of motion of the curved N-body problem on the 3-dimensional manifolds \( {\text{M}}_{k}^{3} \). To achieve this goal, we will define the curved potential function, which also represents the potential of the particle system, introduce and apply Euler’s formula for homogeneous functions to t...

The goal of this chapter is to study polygonal RE in \( {\text{S}}^{ 2} \) and \( {\text{H}}^{ 2} \). Since these manifolds are embedded in \( {\text{R}}^{ 3} \), we will drop the w coordinate from now on and use an xyz frame. Given the fact that the dimension is reduced by one, we will not encounter positive elliptic-elliptic and negative elliptic...

In this chapter we will first introduce the isometries of \( {\text{R}}^{ 4} \) and \( {\text{R}}^{ 3, 1} \) and connect them with the corresponding principal axis theorem. Then we aim to understand how these isometries act in \( {\text{S}}^{ 3} \) and \( {\text{H}}^{ 3} \).

In this chapter we establish criteria for the existence of positive elliptic and elliptic-elliptic as well as negative elliptic, hyperbolic, and elliptic-hyperbolic RE. These criteria will be employed in later chapters to obtain concrete examples of such orbits. The proofs are similar in spirit, but for completeness and future reference we describe...

In 1970, Don Saari conjectured that solutions of the classical N-body problem with constant moment of inertia are relative equilibria. This statement is surprising since one does not expect that such a weak constraint would force the bodies to maintain constant mutual distances all along the motion.

In this chapter we will provide examples of negative RE, one for each type of orbit of this kind: elliptic, hyperbolic, and elliptic-hyperbolic. The first is the Lagrangian RE of equal masses, which is a negative elliptic RE of the 3-body problem in \( {\text{H}}^{ 3} \), the second is the Eulerian orbit of equal masses, which is a negative hyperbo...

we know what kind of rigid-body type orbits to look for in the curved N-body problem for various values of \( N \ge 3 \). Ideal, of course, would be to find them all, but this problem appears to be very difficult, and it might never be completely solved. As a first step towards this (perhaps unreachable) goal, we will show that each type of orbit d...

We consider the motion of point masses given by a natural extension of Newtonian gravitation to spaces of constant positive curvature. Our goal is to explore the spectral stability of tetrahedral orbits of the corresponding 4-body problem in the 2-dimensional case, a situation that can be reduced to studying the motion of the bodies on the unit sph...

We analyze the singularities of the equations of motion and several types of singular solutions of the n-body problem in spaces of positive constant curvature. Apart from collisions, the equations encounter noncollision singularities, which occur when two or more bodies are antipodal. This conclusion leads, on the one hand, to hybrid solution singu...

We provide a class of orbits in the curved N-body problem for which no point that could play the role of the centre of mass is fixed or moves uniformly along a geodesic. This proves that the equations of motion lack centre-of-mass and linear-momentum integrals.

The goal of this chapter is to introduce the concepts we will explore in the rest of this monograph, namely the relative equilibrium solutions, also called relative equilibrium orbits or, simply, relative equilibria (from now on denoted by RE, whether in the singular or the plural form of the noun) of the curved N-body problem. For RE, the particle...

In this chapter we will construct examples of positive elliptic-elliptic RE, i.e. orbits with two elliptic rotations on the sphere \( {\text{S}}^{ 3} \). The first example is that of a 3-body problem in which 3 bodies of equal masses are at the vertices of an equilateral triangle, which has two rotations of the same frequency. The second example is...

In this chapter we will introduce the concept of fixed-point solution, also simply called fixed point (from now on denoted by FP, whether in the singular or the plural form of the noun) of the equations of motion, show that FP exist in \( {\text{S}}^{ 3} \), provide a couple of examples, and finally prove that they don’t show up in \( {\text{H}}^{...

The case \( N = 3 \) presents particular interest in Euclidean space because the equilateral triangle is a RE for any values of the masses, a property discovered by Joseph Louis Lagrange in 1772, We will further show that this is not the case in \( {\text{S}}^{ 2} \) and \( {\text{H}}^{ 2} \), where the positive and negative elliptic Lagrangian RE...

In this chapter we will describe some qualitative dynamical properties for the positive elliptic, positive elliptic-elliptic, negative elliptic, negative hyperbolic, and negative elliptic-hyperbolic RE, under the assumption that they exist. (Examples of such solutions will be given in Part IV for various values of N and of the masses \( {\text{m}}_...

We consider the motion of n point particles of positive masses that interact gravitationally on the 2-dimensional hyperbolic sphere, which has negative constant Gaussian curvature. Using the stereographic projection, we derive the equations of motion of this curved n-body problem in the Poincar\'e disk, where we study the elliptic relative equilibr...

We consider the 3-dimensional gravitational $n$-body problem, $n\ge 2$, in spaces of constant Gaussian curvature $\kappa\ne 0$, i.e.\ on spheres ${\mathbb S}_\kappa^3$, for $\kappa>0$, and on hyperbolic manifolds ${\mathbb H}_\kappa^3$, for $\kappa<0$. Our goal is to define and study relative equilibria, which are orbits whose mutual distances rema...

We extend the Newtonian n-body problem of celestial mechanics to spaces of curvature κ=constant and provide a unified framework for studying the motion. In the 2-dimensional case, we prove the existence of several classes of relative equilibria, including the Lagrangian and Eulerian solutions for any κ≠0 and the hyperbolic rotations for κ<0. These...

In the $2$-dimensional $n$-body problem, $n\ge 3$, in spaces of constant curvature, $\kappa\ne 0$, we study polygonal homographic solutions. We first provide necessary and sufficient conditions for the existence of these orbits and then consider the case of regular polygons. We further use this criterion to show that, for any $n\ge 3$, the regular...

Prologue: Glimpsing the Future xi Chapter 1: Walls of Water: Tsunamis 1 Chapter 2: Land in Upheaval: Earthquakes 21 Chapter 3: Chimneys of Hell: Volcanic Eruptions 42 Chapter 4: Giant Whirlwinds: Hurricanes, Cyclones, and Typhoons 63 Chapter 5: Mutant Seasons: Rapid Climate Change 86 Chapter 6: Earth in Collision: Cosmic Impacts 109 Chapter 7: Econ...

In the 2-dimensional curved 3-body problem, we prove the existence of Lagrangian and Eulerian homographic orbits, and provide their complete classification in the case of equal masses. We also show that the only non-homothetic hyperbolic Eulerian solutions are the hyperbolic Eulerian relative equilibria, a result that proves their instability.

We consider $n$-body problems given by potentials of the form ${\alpha\over r^a}+{\beta\over r^b}$ with $a,b,\alpha,\beta$ 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...

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 $-\beta$, $\beta\ge 2$. For $\beta>2$, the sets of initial conditions leading to collisions/ejections and the one leading to escapes/captures have positive measure. For $\beta>2$ and $\beta\ne 3$, the flow o...

We study singularities of the n-body problem in spaces of constant curvature and generalize certain results due to Painleve, Weierstrass, and Sundman. For positive curvature, some of our proofs use the correspondence between total collision solutions of the original system and their orthogonal projection--a property that offers a new method of appr...

In 1969, D. Saari conjectured that the only solutions of the Newtonian n—body problem that have constant moment of inertia are relative equilibria. For n = 3, there is a computer assisted proof of this conjecture given by R. Moeckel in 2005, [10]. The collinear case was solved the same year by F. Diacu, E. Pérez‐Chavela, and M. Santoprete, [4],...

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 posi...

In 1969, D. Saari conjectured that the only solutions of the Newtonian n—body problem that have constant moment of inertia are relative equilibria. For n = 3, there is a computer assisted proof of this conjecture given by R. Moeckel in 2005, [10]. The collinear case was solved the same year by F. Diacu, E. Pérez‐Chavela, and M. Santoprete, [4], All...

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-...

We study the dynamics of an invariant set of the classical Coulomb atom, which generalizes the one Langmuir proposed for helium in 1921. The n electrons are positioned at the vertices of a regular polygon that changes size homothetically, while the nucleus moves along a line orthogonal through the centre of the polygon. Our main result is that for...

We consider the motion of particles whose interaction is described by a potential of the form \(W(r) = \frac{1}{{|r{|^a}}} + \frac{\mu }{{|r{|^b}}},\mu > 0,0 \leqslant a < b,\) called quasihomogeneous, and discuss the regularization of binary collisions in the two-body and the rectilinear three-body problems for all values of the parameters.

We study the global flow of the anisotropic Manev problem, which describes the planar motion of two bodies under the influence of an anisotropic Newtonian potential with a relativistic correction term. We first find all the heteroclinic orbits between equilibrium solutions. Then we generalize the Poincaré–Melnikov method and use it to prove the exi...

This is an abstract of a paper, which will be published somwhere else. Our goal is to prove some qualitative results related to a generalization of Langmuir's problem, i.e. that of the classical isosceles 3-body problem (two electrons of equal masses and one nucleus, all assumed to be point particles) whose motion is due to a Coulomb force... Note...

We discuss the contributions of Spiru Haret to the problem of the solar system's stability and show their importance relative to the mathematics research fo the late 19th century. We also give a brief survey of the subsequent developments and the consequences of Haret's results.

The anisotropic Manev problem describes the motion of two point masses in an anisotropic space under the influence of a Newtonian force-law with a relativistic correction term. In this paper we prove the existence of a large, open and connected manifold of solutions of the planar anisotropic Manev problem that are uniformly bounded and collisionles...

The anisotropic Manev problem, which lies at the intersection of classical, quantum, and relativity physics, describes the motion of two point masses in an anisotropic space under the influence of a Newtonian force-law with a relativistic correction term. Using an extension of the Poincaré–Melnikov method, we first prove that for weak anisotropy, c...

The anisotropic Manev problem, which lies at the intersection of classical, quantum, and relativity physics, describes the motion of two point masses in an anisotropic space under the influence of a Newtonian force-law with a relativistic correction term. Using an extension of the Poincare'-Melnikov method, we first prove that for weak anisotropy,...

We study the dynamics of particle systems near solution singularities for different types of potentials. Solution singularities appear in finite time, but they can be blown up and removed to infinity with the help of certain transformations, thus creating the framework of a dynamical system. In this paper we present several examples of solution sin...

We consider two‐body problems in which the drag is proportional to the velocity divided by the square of the distance and whose radial and tangential components have distinct coefficients. For all parameters, we study the flow of the system obtained by suitable coordinate and time transformations and draw conclusions about the qualitative behavior...

We consider the Manev potential, given by the sum between the inverse and the inverse square of the distance, in an anisotropic space, i.e., such that the force acts differently in each direction. Using McGehee coordinates, we blow up the collision singularity, paste a collision manifold to the phase space, study the flow on and near the collision...

We consider the Gylden problem-a perturbation of the Kepler problem via an explicit function of time. For certain general classes of planar periodic perturbations, after proving a Poincare'-Melnikov-type criterion, we find a manifold of orbits in which the dynamics is given by the shift automorphism on the set of bi-infinite sequences with infinite...

The Manev problem (a two-body problem given by a potential of the form A/r+B/r2, where r is the distance between particles and A,B are positive constants) comprises several important physical models, having its roots in research done by Isaac Newton. We provide its analytic solution, then completely describe its global flow using McGehee coordinate...

We study collision and ejection orbits of 3-particle systems having the potentialW=U+V, whereUandVare homogeneous functions of degree −aand −b, respectively, with 1⩽a<b. We show that forb≠2, collision and ejection orbits tend to form asymptotically a central configuration. For the caseb=2, which corresponds to Maneff's gravitational law, we find a...

We consider the 3-body problem with the Mücket-Treder post-Newtonian gravitational law. We show that for certain cases singularities cannot occur, while in the other cases triple collision solutions tend to form Newtonian central configurations near collision.

Manev’s gravitational law, given by a potential function of the form U(ρ)+ν/ρ=μ/ρ 2 , where ρ is the distance between particles and ν, μ are suitably chosen positive constants, is a fairly good substitute of relativity within the frame of classical mechanics. We first obtain exact formulae for the trajectory and for the nodal period. Then we procee...

A large class of symmetry solutions of the Newtonian n-body problem cannot end in a noncollision singularity nor expand faster than any constant multiple of time.

Maneff’s gravitational law explains, with a very good approximation, the perihelion advance of the inner planets as well as the orbit of the Moon. Here the invariant set of planar isosceles solutions of the three‐body problem for Maneff’s model is studied. The application of Maneff’s law in atomic physics provides, in the case of the isosceles prob...

This is a story about celestial mechanics and mathematics and about a question older than Bieberbach’s conjecture; a question that died close to its 100th birthday but which – like any good question – left behind it many other unanswered questions as well as a universe of intellectual achievements.

The goal of this paper is to bring to the attention of the astronomic community the recent mathematical results of Xia on the existence of the Arnold diffsion in the 3 - body problem and to speculate on its significance for the astronomy of the solar system. We bring evidence that thus result supports the idea of the existence of small celestial bo...

We study the central force problem involving a particle of unit mass orbiting a centre while subjected to a post-Newtonian gravitational force proportional to r-2 (1 + __ 1n r), where r is the distance to the centre and |__|

For a set of masses having positive measure, excepting eventually a negligible set of initial conditions, every noncollinear bounded solution of the planar three-body problem that has a syzygy configuration encounters an infinity of such configurations. Along a noncollinear syzygy solution, the set of syzygy configuration instants is discrete.

For a set of masses having positive measure, excepting eventually a negligible set of initial conditions, every noncollinear bounded solution of the planar three-body problem that has a syzygy configuration encounters an infinity of such configurations. Along a noncollinear syzygy solution, the set of syzygy configuration instants is discrete.

We show that every planar isosceles solution of the three-body problem encounters a collision of the symmetric particles, either forwards or backwards in time. Regularizing analytically this collision, the solution has at least a syzygy configuration and/or leads to a total collapse. Some further simple results support the intuitive image on the ta...

We prove that if a planar solution of the n-body problem has a symmetry axis, fixed with respect to the considered frame, and the center of mass of the particle system lies on this axis during the motion, then the symmetric masses must be equal. We also show that the set of initial conditions leading to symmetric solutions has measure zero and is n...

We study solutions leading to a simultaneous total collapse in the n-body problem with generalized attraction law given by the inverse (α+1)-power of the distance, α>0. We generalize a result due to Weierstrass regarding the angular momentum constant and prove further that, for α=2, the collision instant can be computed explicitly as a function of...

Relations between the rectilinear, collinear and syzygy solutions of the N-body problem are first pointed out. It is shown that, along a solution, the set of the non-collinear syzygy configuration instants is formed by isolated points. Then we restrict the study to the planar 3-body problem and prove that for Dirichlet-stable solutions, a non-syzyg...

It is proved that if a non-collinear motion of the four body problem has a symmetry axis (or plane), then the center of mass lies on this axis (plane) and the symmetric masses are equal. We also remark that this result is true for the generalized attraction law given by the inverse (α+1)-power of the distance, with α > 0.

It is shown that in then-body problem with generalized attraction law, the sets of initial conditions which lead to Wintner's collinear, respectively flat, motion are nowhere dense relative to the set of initial conditions that define solutions inR 3.

It is shown that in the n-body problem with generalized attraction law (inverse (α + l )-power of the distance, α > 0) the set of initial conditions which lead to collinear motion is of Lebesgue measure zero and nowhere dense with respect to the set of initial conditions that define solutions in R3 or R2. Es wird gezeigt, daß beim n-Körperproblem m...

We consider the simultaneous collision of n bodies in the general N-body problem, 2 ≤ n ≤ N, in the k-dimensional Euclidian space and eliminate the singularities which arise from that collision in the equations of motion by suitable coordinates and time transformations. The singularity of that solution is removed by throwing the collision instant t...

A refinement of the Sundman inequality containing the error term is generalized for the inverse (alpha+1)-force law, alpha > 0, in the k-dimensional Euclidean space, for the n-body problem, n >= 2.