Geometry of almost-conserved quantities in symplectic maps

Tim Zolkin, Sergei Nagaitsev, Ivan Morozov and Sergei Kladov

Chaos, Solitons & Fractals, 2026, vol. 208, issue P1

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.

Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:208:y:2026:i:p1:s0960077926002006

DOI: 10.1016/j.chaos.2026.118059

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