Construction of approximate invariants for non-integrable Hamiltonian systems

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.

💡 Research Summary

The paper presents a systematic method for constructing high‑order polynomial approximate invariants (AIs) in non‑integrable Hamiltonian systems and demonstrates its practical application to modern ring‑based particle accelerators. The authors start by representing the one‑turn map of a periodic Hamiltonian system as a square matrix M that transforms an extended phase‑space vector Z(Ω), which contains all monomials of the canonical coordinates up to order Ω. An AI of order Ω is defined as a linear combination W(Ω)=VᵀZ(Ω) that remains unchanged after one turn, leading to the eigenvalue condition V = MᵀV with eigenvalue 1. Direct eigen‑analysis of Mᵀ fails for Ω ≥ 3 because the coefficients of higher‑order terms grow factorially, causing the eigenvectors to be dominated by the highest‑order monomials.

To overcome this, the authors adopt an iterative, order‑by‑order construction. They first extract the linear Courant‑Snyder invariants from the 2‑nd‑order (14 × 14) block of Mᵀ. Using the block‑triangular structure of the higher‑order matrices—where the upper‑right blocks are zero—they derive a simple linear system for the next‑order coefficients: V(Ω) = (I − m_Ω)⁻¹ m_Ω‑1 V(Ω‑1). The matrix (I − m_Ω) is invertible as long as the fundamental tune is chosen off resonance, a condition typically satisfied in accelerator lattices. By repeating this procedure, cubic, quartic, and quintic AIs are obtained explicitly, with their coefficients verified against the one‑turn transformation to machine‑precision (differences ≈ 10⁻¹³).

The method is applied to the NSLS‑II storage ring. The authors compute AIs up to fifth order and evaluate them using a 128‑turn symplectic tracking simulation for a set of initial conditions. As the order increases, the fluctuations of the AIs (defined as σ_W/|⟨W⟩|) decrease, indicating improved approximation, but beyond fifth order the fluctuations begin to rise, suggesting either accumulated numerical error or genuine breakdown of the approximate tori. The AI fluctuation is then proposed as a chaos indicator. It is compared with frequency‑map analysis (FMA), Shannon entropy, and forward‑reversal integration. All four metrics show a similar transition from regular to chaotic motion with increasing amplitude, yet the AI fluctuation offers a direct, analytically defined measure that is computationally cheap and readily incorporated into optimization objectives.

Exploiting this property, the authors perform a lattice optimization by tuning a set of harmonic sextupoles to minimize the fifth‑order AI fluctuations for a specific seed trajectory (x = 15 mm, y = 3.7 mm). After optimization, on‑ and off‑momentum dynamic apertures are recomputed with 2048‑turn tracking. The resulting apertures are comparable to the baseline design, while the AI fluctuations are substantially reduced, demonstrating that minimizing AI fluctuations is a more efficient route to enlarging the stable region than conventional multi‑particle tracking‑based optimizations. The entire optimization completes within a few hours on a single‑core workstation.

In conclusion, the paper delivers three major contributions: (1) an algorithmic framework for constructing high‑order polynomial approximate invariants by leveraging the block‑zero structure of the one‑turn map; (2) the identification of AI fluctuation as a quantitative chaos indicator that correlates with traditional metrics; and (3) a practical demonstration that minimizing AI fluctuations can be used as an objective function to enlarge the dynamic aperture of a modern accelerator. The approach is general and can be extended to other non‑integrable Hamiltonian systems where long‑term stability and control are critical.