Geometry of almost-conserved quantities in symplectic maps

Abstract

Noether’s theorem, which connects continuous symmetries to exact conservation laws, remains one of the most fundamental principles in physics and dynamical systems. In this work, we draw a conceptual parallel between two paradigms: the emergence of exact invariants from continuous symmetries, and the appearance of approximate invariants from discrete symmetries associated with reversibility in symplectic maps. We demonstrate that by constructing approximating functions that preserve these discrete symmetries order by order, one can systematically uncover hidden structures, closely echoing Noether’s framework. The resulting unified function, constructed from the map coefficients, produces phase portraits, rotation numbers, and tune footprints that closely agree with numerical tracking over wide parameter ranges, thereby offering a compact representation of near-integrable dynamics.

Using the symmetric form of the map as a flexible test case, we benchmark the perturbative construction against established techniques, including the Lie algebra method for twist coefficients. To resolve the inherent ambiguity in the perturbation series, we introduce an averaging procedure that naturally leads to a resonant theory — capable of treating rational rotation numbers and small-denominator divergences. This enables an accurate and structured description of low-order resonances, including singular and non-singular features in the quadratic and cubic Hénon maps. We also explore the recovery of exact invariants in integrable cases of McMillan and Suris mappings, showing that the method reproduces the correct behavior when such structure is present. Comparisons with the Square Matrix method reveal that while both approaches satisfy local constraints, our technique provides greater accuracy and robustness in resonant and strongly nonlinear regimes. The approach is systematic, requiring only linear algebra and integrals of elementary functions, yet it yields results in striking agreement with both theory and numerical experiment.