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Nondeterministic particle systems

Andreas Knauf, Manuel Quaschner

Abstract

We consider systems of n particles that move with constant velocity between collisions. Their total momentum but not necessarily their kinetic energy is preserved at collisions. As there are no further constraints, these systems are nondeterministic. In particular we examine trajectories with infinitely many collisions.

Nondeterministic particle systems

Abstract

We consider systems of n particles that move with constant velocity between collisions. Their total momentum but not necessarily their kinetic energy is preserved at collisions. As there are no further constraints, these systems are nondeterministic. In particular we examine trajectories with infinitely many collisions.

Paper Structure

This paper contains 13 sections, 20 theorems, 82 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Phase Flow
  4. Cluster Decomposition
  5. Radial coordinates
  6. Nondeterministic systems
  7. Definition and elementary properties
  8. Moment of inertia
  9. Exponentially expanding systems
  10. Closedness of nondeterministic particle systems
  11. Lagrangian relations for the non-deterministic system
  12. Scattering by a single subspace
  13. Scattering by several subspaces

Key Result

Lemma 2.3

The vector space automorphisms are symplectic w.r.t. the natural symplectic forms on these cotangent bundles.

Figures (1)

  • Figure 4.1: Solutions in the composition $L_{{\cal C}_2,{\cal C}_3;E_2,E_3} \circ L_{{\cal C}_1,{\cal C}_2;E_1,E_2}$ of Lagrangian relations that intersect $\Delta_{{\cal C}_3}$ (red), respectively do not (green)

Theorems & Definitions (40)

  • Example 1.1: Celestial mechanics
  • Definition 2.1
  • Example 2.2: Collision Subspaces
  • Lemma 2.3
  • Example 2.4: Pair Clusters
  • Remark 2.5: partition of configuration space
  • Definition 3.1
  • Lemma 3.3
  • Lemma 3.4: projections of nondeterministic trajectories
  • Remark 3.5: extension of nondeterministic trajectories
  • ...and 30 more