Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 114, 24 pages      arXiv:2009.10417      https://doi.org/10.3842/SIGMA.2024.114

Real Forms of Holomorphic Hamiltonian Systems

Philip Arathoon a and Marine Fontaine b
a) Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA
b) Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

Received June 12, 2024, in final form December 10, 2024; Published online December 21, 2024

Abstract
By complexifying a Hamiltonian system, one obtains dynamics on a holomorphic symplectic manifold. To invert this construction, we present a theory of real forms which not only recovers the original system but also yields different real Hamiltonian systems which share the same complexification. This provides a notion of real forms for holomorphic Hamiltonian systems analogous to that of real forms for complex Lie algebras. Our main result is that the complexification of any analytic mechanical system on a Grassmannian admits a real form on a compact symplectic manifold. This produces a 'unitary trick' for Hamiltonian systems which curiously requires an essential use of hyperkähler geometry. We demonstrate this result by finding compact real forms for the simple pendulum, the spherical pendulum, and the rigid body.

Key words: Hamiltonian dynamics; integrable systems; hyperkähler geometry.

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