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Jointly Equivariant Dynamics for Interacting Particles

Alain Ajami, Jean-Paul Gauthier, Francesco Rossi

Abstract

Let a finite set of interacting particles be given, together with a symmetry Lie group $G$. Here we describe all possible dynamics that are jointly equivariant with respect to the action of $G$. This is relevant e.g., when one aims to describe collective dynamics that are independent of any coordinate change or external influence. We particularize the results to some key examples, i.e. for the most basic low dimensional symmetries that appear in collective dynamics on manifolds.

Jointly Equivariant Dynamics for Interacting Particles

Abstract

Let a finite set of interacting particles be given, together with a symmetry Lie group . Here we describe all possible dynamics that are jointly equivariant with respect to the action of . This is relevant e.g., when one aims to describe collective dynamics that are independent of any coordinate change or external influence. We particularize the results to some key examples, i.e. for the most basic low dimensional symmetries that appear in collective dynamics on manifolds.
Paper Structure (26 sections, 36 theorems, 104 equations, 5 figures)

This paper contains 26 sections, 36 theorems, 104 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Definitions and General Results
  3. Equivariance: $G$-spaces and radial functions
  4. Population Dynamics
  5. Families of Population Dynamics with linear group actions
  6. Time reparametrization
  7. Permutation-Equivariant Population Dynamics with linear group action
  8. Translation equivariance in Euclidean and pseudo-Euclidean spaces
  9. Population Dynamics on Lie groups
  10. Gradient flow Population Dynamics
  11. Population Dynamics on the Euclidean plane
  12. Gradient flows for 2 particles
  13. Permutation-equivariant population dynamics
  14. Population Dynamics on the relativistic spaces
  15. Population Dynamics on the relativistic line
  16. ...and 11 more sections

Key Result

Proposition 2

The vector field $F$ is $G$-equivariant if and only if $[F,l(x)]=0$ for all $l\in\mathfrak{L}$, $x\in X$.

Figures (5)

  • Figure 1: Examples A (1 to 12).
  • Figure 2: Examples B (1 to 6). Variables $T$ above, $x$ below; Variables 1 (blue) and 2 (orange).
  • Figure 3: Examples C (1 to 4).
  • Figure 4: Examples on $S^2$ with $N=1$ and $N=2$.
  • Figure 5: A Controlled Population Dynamics for the unicycle with $N=2$.

Theorems & Definitions (93)

  • Definition 1: $G$-space
  • Proposition 2
  • proof
  • Definition 3
  • Proposition 4
  • proof
  • Definition 5
  • Definition 6
  • Corollary 7
  • Remark 8
  • ...and 83 more