Construction of approximate invariants for non-integrable Hamiltonian systems

TL;DR

This paper introduces a method to construct high-order polynomial approximate invariants for non-integrable Hamiltonian systems, aiding in chaos detection and stability enhancement in particle accelerators.

Contribution

The paper presents a novel iterative approach to build approximate invariants for complex Hamiltonian systems, applicable to modern accelerators.

Findings

AI fluctuations correlate with chaos levels

Minimizing AI fluctuations enlarges stable motion regions

Method effectively detects and controls chaos in accelerators

Abstract

We present a method to construct high-order polynomial approximate invariants (AI) for non-integrable Hamiltonian dynamical systems, and apply it to modern ring-based particle accelerators. Taking advantage of a special property of one-turn transformation maps in the form of a square matrix, AIs can be constructed order-by-order iteratively. Evaluating AI with simulation data, we observe that AI's fluctuation is actually a measure of chaos. Through minimizing the fluctuations with control knobs in accelerators, the stable region of long-term motions could be enlarged.