Birkhoff normal forms, Dirac brackets and symplectic reduction

TL;DR

This paper develops a Dirac-bracket approach to normal forms in constrained Hamiltonian systems, relating it to symplectic reduction, and applies it to study Birkhoff normal forms near relative equilibria, including singular cases.

Contribution

It introduces a Dirac-bracket framework for normal forms on momentum levels and connects it to symplectic reduction, enabling analysis of singular reduced spaces.

Findings

Dirac brackets coincide with evolution on local slices for invariant Hamiltonians.

Birkhoff normal forms can be constructed on $J^{-1}(a0)$ and descend to reduced strata.

Application to the double spherical pendulum demonstrates the method.

Abstract

Dirac brackets are widely used to study constrained Hamiltonian dynamics. In this paper we develop a Dirac-bracket approach to normal forms on momentum levels and relate it to symplectic reduction in the cases where reduction yields a (stratified) symplectic quotient. We consider a proper Hamiltonian $G$-action on a symplectic manifold $(M,\omega)$ with an equivariant momentum map $J$. We fix $\mu \in \mathfrak g^*$and work on $J^{-1}(\mu)$. For $G$-invariant Hamiltonians whose induced vector field on $J^{-1}(\mu)$ is tangent to a local $G_\mu$-slice, we show that the induced evolution on $J^{-1}(\mu)$ coincides with that defined by the Dirac bracket on a local second-class slice, and descends to the corresponding symplectic stratum of $J^{-1}(\mu)/G_\mu$. As a main application we study Birkhoff normal forms near a relative equilibrium. When the quadratic part of a symmetric Hamiltonian is tangent to a local $G_\mu$-slice, a Birkhoff normal form can be constructed entirely on the manifold $J^{-1}(\mu)$, and it descends to a Birkhoff normal form for the reduced dynamics on the corresponding stratum, even when the reduced space is singular. We show that for a class of simple mechanical systems this condition holds automatically at a relative equilibrium. We illustrate the method on the double spherical pendulum. Finally, we relate our results to Moser's constrained dynamics by identifying Moser's constrained vector field with the Dirac Hamiltonian vector field. We show that, if the reduced Hamiltonian is near-integrable on a stratum, then its pullback to the symplectic slice is near-integrable with respect to the Dirac bracket, and vice versa. In particular, this provides a practical route to KAM-type results for the constrained dynamics.