Real Forms of Holomorphic Hamiltonian Systems

TL;DR

This paper develops a theory of real forms for holomorphic Hamiltonian systems, allowing the recovery of original systems and the construction of new real systems sharing the same complexification, with applications to classical mechanical systems.

Contribution

It introduces a novel framework for real forms of holomorphic Hamiltonian systems, connecting complexification with hyperk"ahler geometry and providing explicit examples.

Findings

Any analytic mechanical system on a Grassmannian admits a real form on a compact symplectic manifold.

The theory yields a 'unitary trick' for Hamiltonian systems using hyperk"ahler geometry.

Explicit real forms are found for the simple pendulum, spherical pendulum, and rigid body.

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 hyperk\"ahler geometry. We demonstrate this result by finding compact real forms for the simple pendulum, the spherical pendulum, and the rigid body.