Geometry of almost-conserved quantities in symplectic maps: Approximate invariants in nonlinear accelerator systems

We present a perturbative method for constructing approximate invariants of motion directly from the equations of discrete-time symplectic systems. This framework offers a natural nonlinear extension of the classic Courant-Snyder (CS) theory for systems with 1 degree of freedom—a foundational cornerstone in accelerator physics now spanning seven decades and historically focused on linear phenomena. The original CS formalism emerged under conditions where nonlinearities were weak, design goals favored linear motion, and analytical tools—such as the Kolmogorov-Arnold-Moser theory—had not yet been fully developed. While various normal-form methods have been proposed to treat near-integrable dynamics, the approach introduced here stands out for its conceptual transparency, minimal computational overhead, and direct applicability to realistic systems. We demonstrate its power and versatility by applying it to several operational accelerator configurations at the Fermi National Accelerator Laboratory (Fermilab), illustrating how the method enables fast, interpretable diagnostics of nonlinear behavior across a broad range of machine conditions.