Nonexistence of Integrable Nonlinear Magnetic Fields with Invariants Quadratic in Momenta

TL;DR

This paper proves that integrable nonlinear magnetic fields with invariants quadratic in momenta do not exist under strict physical constraints, limiting the types of integrable systems applicable in advanced accelerator design.

Contribution

It demonstrates the nonexistence of certain integrable magnetic fields with quadratic invariants and conjectures the absence of higher-degree invariants, highlighting fundamental limitations.

Findings

No such integrable systems with quadratic invariants exist under Maxwell constraints.

Many solutions exist without Maxwell constraints, including a large family of integrable Hamiltonians.

Only linear invariants are compatible with solenoidal magnetic fields under strict conditions.

Abstract

Nonlinear, completely integrable Hamiltonian systems that serve as blueprints for novel particle accelerators at the intensity frontier are promising avenues for research, as Fermilab's Integrable Optics Test Accelerator (IOTA) example clearly illustrates. Here, we show that only very limited generalizations are possible when no approximations in the underlying Hamiltonian or Maxwell equations are allowed, as was the case for IOTA. Specifically, no such systems exist with invariants quadratic in the momenta, precluding straightforward generalization of the Courant-Snyder theory of linear integrable systems in beam physics. We also conjecture that no such systems exist with invariants of higher degree in the momenta. This leaves solenoidal magnetic fields, including their nonlinear fringe fields, as the only completely integrable static magnetic fields, albeit with invariants that are linear in the momenta. The difficulties come from enforcing Maxwell equations; without constraints, we show that there are many solutions. In particular, we discover a previously unknown large family of integrable Hamiltonians.