Geometry of Almost-Conserved Quantities in Symplectic Maps. Part I: Perturbation Theory

TL;DR

This paper develops a perturbation theory framework for symplectic maps that constructs approximate invariants preserving discrete symmetries, providing insights into near-integrable behavior and resonances.

Contribution

It introduces a systematic method to build approximate invariants in symplectic maps using symmetry-preserving functions, extending Noether's ideas to discrete symmetries and resonant cases.

Findings

Benchmarking against established techniques shows high accuracy.

Resonant behaviors are effectively captured with the averaging procedure.

Method is simple, requiring only linear algebra and elementary integrals.

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 functions serve not only as diagnostic tools but also as compact representations of near-integrable behavior. The first article establishes the formal foundations of the method. 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 H\'enon maps. 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. We conclude by outlining extensions to more general maps and discussing implications for stability estimates in practical systems such as particle accelerators.