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Uniqueness of hyperbolic Busemann functions in the Newtonian N-body problem

Ezequiel Maderna, Andrea Venturelli

TL;DR

This work shows that in the Newtonian $N$-body problem, hyperbolic rays with the same limit shape define the same Busemann function up to an additive constant, linking asymptotic geometry to weak KAM theory. It develops the notion of hyperbolic viscosity solutions of the Hamilton–Jacobi equation at positive energy, showing uniqueness up to constants and connecting calibrating curves to hyperbolic motions. A key cone theorem provides local uniqueness of hyperbolic motions and yields differentiability of Busemann functions in a cutted cone, which then extends globally, implying that every hyperbolic motion is eventually a minimizer for the Jacobi–Maupertuis metric. The limit shape map is proven to be $\mathcal{C}^1$, with its derivative expressed via Jacobi fields, tying variational stability to asymptotic dynamics. Collectively, these results imply generic (almost-everywhere) uniqueness of geodesic rays with a given limit shape and a well-defined Gromov boundary structure for the Jacobi–Maupertuis metric at positive energy levels.

Abstract

For the N-body problem we prove that any two hyperbolic rays having the same limit shape define the same Busemann function. We localize a region of differentiability for these functions, of which we know that they are viscosity solutions of the stationary Hamilton-Jacobi equation. As a first corollary, we deduce that every hyperbolic motion of the $N$-body problem must become, after some time, a calibrating curve for the Busemann function associated to its limit shape. This implies that every hyperbolic motionof the $N$-body problem is eventually a minimizer, that is, it must contain a geodesic ray of the Jacobi-Maupertuis metric. Since the viscosity solutions of the Hamilton-Jacobi equation are almost everywhere differentiable, we also deduce the generic uniqueness of geodesic rays with a given limit shape without collisions. That is to say, if the limit shape is given, then for almost every initial configuration the geodesic ray is unique.

Uniqueness of hyperbolic Busemann functions in the Newtonian N-body problem

TL;DR

This work shows that in the Newtonian -body problem, hyperbolic rays with the same limit shape define the same Busemann function up to an additive constant, linking asymptotic geometry to weak KAM theory. It develops the notion of hyperbolic viscosity solutions of the Hamilton–Jacobi equation at positive energy, showing uniqueness up to constants and connecting calibrating curves to hyperbolic motions. A key cone theorem provides local uniqueness of hyperbolic motions and yields differentiability of Busemann functions in a cutted cone, which then extends globally, implying that every hyperbolic motion is eventually a minimizer for the Jacobi–Maupertuis metric. The limit shape map is proven to be , with its derivative expressed via Jacobi fields, tying variational stability to asymptotic dynamics. Collectively, these results imply generic (almost-everywhere) uniqueness of geodesic rays with a given limit shape and a well-defined Gromov boundary structure for the Jacobi–Maupertuis metric at positive energy levels.

Abstract

For the N-body problem we prove that any two hyperbolic rays having the same limit shape define the same Busemann function. We localize a region of differentiability for these functions, of which we know that they are viscosity solutions of the stationary Hamilton-Jacobi equation. As a first corollary, we deduce that every hyperbolic motion of the -body problem must become, after some time, a calibrating curve for the Busemann function associated to its limit shape. This implies that every hyperbolic motionof the -body problem is eventually a minimizer, that is, it must contain a geodesic ray of the Jacobi-Maupertuis metric. Since the viscosity solutions of the Hamilton-Jacobi equation are almost everywhere differentiable, we also deduce the generic uniqueness of geodesic rays with a given limit shape without collisions. That is to say, if the limit shape is given, then for almost every initial configuration the geodesic ray is unique.

Paper Structure

This paper contains 18 sections, 20 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Hyperbolic motions
  3. Geodesic rays and Busemann functions
  4. Hyperbolic viscosity solutions
  5. Hyperbolic solutions to the Hamilton-Jacobi equation
  6. Uniqueness of the hyperbolic solution
  7. Proof of the main theorems and some consequences
  8. Theorem of the cone
  9. Proof of the main results
  10. Corollaries of the main theorems
  11. The limit shape map
  12. The Jacobi equation
  13. A lemma on the growth of Jacobi fields
  14. Regularity of the limit shape map
  15. Uniqueness of hyperbolic motions
  16. ...and 3 more sections

Key Result

Theorem 1.1

Given $h>0$ and two hyperbolic geodesic rays $\gamma_1(t)$ and $\gamma_2(t)$, the following are equivalent.

Theorems & Definitions (70)

  • Definition 1.1: The Jacobi-Maupertuis distance
  • Remark 1.1
  • Definition 1.2: Busemann functions
  • Definition 1.3
  • Theorem 1.1
  • Remark 1.2: Chazy's estimate
  • Remark 1.3
  • Definition 1.4: Busemann function associated with a configuration
  • Theorem 1.2
  • Remark 1.4
  • ...and 60 more