The Birkhoff Normal Form through the Lens of Representation Theory

    2025, Volume 30, Number 6, pp.  952–961

    Author(s): Montgomery R.

    We derive the simplest version of the finite-order Birkhoff normal forms [BNFs], that for area-preserving maps of the plane, using the finite-dimensional representation theory for the group of linear area-preserving maps of that plane and of its circle subgroup.We describe our motivation: the utility of understanding the 3rd-order BNF to obtain KAM stability for non-trivial periodic orbits which arise in celestial mechanics.
    Keywords: Birkhoff normal form, area-preserving map, finite-dimensional representation theory of $SL(2, \mathbb{R})$, KAM theory, homological equations
    Citation: Montgomery R., The Birkhoff Normal Form through the Lens of Representation Theory, Regular and Chaotic Dynamics, 2025, Volume 30, Number 6, pp. 952–961
    DOI: 10.1134/S1560354725520016



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