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Über dieses Buch

The guiding light of this monograph is a question easy to understand but difficult to answer: {What is the shape of the universe? In other words, how do we measure the shortest distance between two points of the physical space? Should we follow a straight line, as on a flat table, fly along a circle, as between Paris and New York, or take some other path, and if so, what would that path look like? If you accept that the model proposed here, which assumes a gravitational law extended to a universe of constant curvature, is a good approximation of the physical reality (and I will later outline a few arguments in this direction), then we can answer the above question for distances comparable to those of our solar system. More precisely, this monograph provides a mathematical proof that, for distances of the order of 10 AU, space is Euclidean. This result is, of course, not surprising for such small cosmic scales. Physicists take the flatness of space for granted in regions of that size. But it is good to finally have a mathematical confirmation in this sense. Our main goals, however, are mathematical. We will shed some light on the dynamics of N point masses that move in spaces of non-zero constant curvature according to an attraction law that naturally extends classical Newtonian gravitation beyond the flat (Euclidean) space. This extension is given by the cotangent potential, proposed by the German mathematician Ernest Schering in 1870. He was the first to obtain this analytic expression of a law suggested decades earlier for a 2-body problem in hyperbolic space by Janos Bolyai and, independently, by Nikolai Lobachevsky. As Newton's idea of gravitation was to introduce a force inversely proportional to the area of a sphere the same radius as the Euclidean distance between the bodies, Bolyai and Lobachevsky thought of a similar definition using the hyperbolic distance in hyperbolic space. The recent generalization we gave to the cotangent potential to any number N of bodies, led to the discovery of some interesting properties. This new research reveals certain connections among at least five branches of mathematics: classical dynamics, non-Euclidean geometry, geometric topology, Lie groups, and the theory of polytopes.



Chapter 1. Introduction

In this introductory chapter, we provide the motivation that led us to this research, define the problem, explain its importance, outline its history, and present the structure of the monograph.
Florin Diacu

Background and Equations of Motion


Chapter 2. Preliminary Developments

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 attributed to Hendrik Lorentz, but actually due to Karl Weierstrass.
Florin Diacu

Chapter 3. Equations of motion

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 the curved potential function, describe the variational method of constrained Lagrangian dynamics, and write down the Euler–Lagrange equations with constraints. After deriving the equations of motion of the curved N-body problem, we will prove that their study can be reduced, by suitable coordinate and time-rescaling transformations, to the unit manifold \( {\text{M}}^{3} \). Finally, we will show that the equations of motion can be put in Hamiltonian form and will find their first integrals.
Florin Diacu

Isometries and Relative Equilibria


Chapter 4. Isometric Rotations

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 \( {\text{H}}^{ 3} \).
Florin Diacu

Chapter 5. Relative Equilibria (RE)

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 system behaves like a rigid body, i.e. all the mutual distances between the point masses remain constant during the motion.
Florin Diacu

Chapter 6. Fixed Points (FP)

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 \( {\text{H}}^{ 3} \) and in hemispheres of \( {\text{S}}^{ 3} \), provided that at least one body is not on the boundary of the hemisphere.
Florin Diacu

Criteria and Qualitative Behavior


Chapter 7. Existence Criteria

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 them all since the specifics differ in each case. We close this chapter by showing that negative parabolic RE do not exist in the curved N-body problem.
Florin Diacu

Chapter 8. Qualitative Behavior

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 \( {\text{m}}_{ 1} , {\text{m}}_{ 2} ,\ldots , {\text{m}}_{\text{N}} { > 0} . \)) For this purpose we will also provide some geometric-topologic considerations about \( {\text{S}}_{ 3} \) and \( {\text{H}}_{ 3} \).
Florin Diacu



Chapter 9. Positive Elliptic RE

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 described in the above criteria and theorems exists for some values of \( N \ge 3 \) and \( m_{1} ,m_{2} , \ldots ,m_{N} > 0 \).
Florin Diacu

Chapter 10. Positive Elliptic-Elliptic RE

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 that of a 4-body problem in which 4 equal masses are at the vertices of a regular tetrahedron, which has two rotations of the same frequency. The third example is that of a 5-body problem in which 5 equal masses lying at the vertices of a pentatope have two rotations of same-size frequencies. This is the only regular polytope that allows RE, because the five other existing regular polytopes of \( {\text{R}}^{ 4} \) have antipodal vertices, so they introduce singularities.
Florin Diacu

Chapter 11. Negative RE

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 hyperbolic RE of the 3-body problem in \( {\text{H}}^{ 3} \), and the third is an elliptic-hyperbolic orbit that combines the previous two examples in the sense that it inherits their rotations.
Florin Diacu

The 2-dimensional case


Chapter 12. Polygonal RE

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-hyperbolic RE anymore. So the only orbits we will deal with from now on are the positive and negative elliptic as well as the negative hyperbolic RE.
Florin Diacu

Chapter 13. Lagrangian and Eulerian RE

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 exist only if the masses are equal. This conclusion provides a first step towards understanding with the help of these equations whether space is Euclidean for distances of the order of 10 AU because Lagrangian orbits of unequal masses show up in our solar system, as for example the approximate equilateral triangle formed by the Sun, Jupiter, and the Trojan/Greek asteroids.
Florin Diacu

Chapter 14. Saari’s Conjecture

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.
Florin Diacu


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