# Induced dynamics

## TL;DR

This paper introduces induced dynamical systems based on the zeros of a function with evolving parameters, revealing complex particle interactions that mimic quantum effects while maintaining Hamiltonian and integrable properties.

## Contribution

It defines a new class of induced dynamical systems that model particle collisions, bound states, and creation/annihilation phenomena within a Hamiltonian framework.

## Key findings

- Demonstrates nontrivial particle collisions in induced systems
- Shows induced systems can model quantum-like effects
- Establishes Hamiltonian and Liouville integrability of these systems

## Abstract

Induced dynamics is defined as dynamics of real zeros with respect to $x$ of equation $f(q_1-x,\ldots,q_N-x,p_1,\ldots,p_N)=0$, where $f$ is a function, and $q_i$ and $p_j$ are canonical variables obeying some (free) evolution. Identifying zero level lines with the world lines of particles, we show that the resulting dynamical system demonstrates highly nontrivial collisions of particles. In particular, induced dynamical systems can describe such ``quantum'' effects as bound states and creation/annihilation of particles, both in nonrelativistic and relativistic cases. On the other side, induced dynamical systems inherit properties of the $(p,q)$-systems being Hamiltonian and Liouville integrable.