In Memory of Alexey V. Borisov (on his 60th Birthday): Part I (Issue Editors: Ivan Mamaev and Iskander Taimanov)

The Birkhoff Normal Form through the Lens of Representation Theory

Richard Montgomery

Dept. of Mathematics, UCSC, 1156 High Street, 95064 Santa Cruz, CA

Abstract: 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

Received: 28.04.2025
Accepted: 04.06.2025

Language: English

DOI: 10.1134/S1560354725520016