Birkhoff normal forms, Dirac brackets and symplectic reduction

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 $(M,ω)$ with an equivariant momentum map $J$. We fix $μ\in \mathfrak g^*$and work on $J^{-1}(μ)$. For $G$-invariant Hamiltonians whose induced vector field on $J^{-1}(μ)$ is tangent to a local $G_μ$-slice, we show that the induced evolution on $J^{-1}(μ)$ coincides with that defined by the Dirac bracket on a local second-class slice, and descends to the corresponding symplectic stratum of $J^{-1}(μ)/G_μ$. 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 $G_μ$-slice, a Birkhoff normal form can be constructed entirely on the manifold $J^{-1}(μ)$, 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.

Figures (5)

• Figure 2.1: Illustration of Dirac's distinction between first-class and second-class constraints. On the left, the first-class constraint arises from the cotangent-lifted $SO(3)$-action on $T^*\mathds{R}^3$ and is given by the zero level of the momentum map, $J(q,p)=q\times p=0$. Its Hamiltonian vector fields are tangent to the $SO(3)$-orbits, so the constrained manifold retains the symmetry directions and is naturally presymplectic. On the right, the second-class constraint describes a particle constrained to the sphere $S_r^2$, with $\phi_1(q,p)=|q|^2-r^2=0$ fixing the position on the sphere and $\phi_2(q,p)=q\cdot p=0$ removing the normal momentum component. In contrast with the first-class case, these constraints cut out a genuine symplectic constrained phase space, whose dynamics is described by the Dirac bracket.
• Figure 2.2: Local slice for a proper $K$-action. The slice $S$ passes through $x_0$ and is transverse to the orbit $K\cdot x_0$. Its saturation $K\cdot S$ is an open neighborhood of $x_0$, locally modeled by the associated bundle $K\times_{K_{x_0}} S$. In particular, $T_xN = T_x(K\cdot x)\oplus T_xS$ for $x\in S$ near $x_0$.
• Figure 4.1: Geometry of the second-class constraints. On the left, the momentum level $J_\mu$ (in blue), a neighborhood $Z\subset M$ of the relative equilibrium $x_0$ (in grey), and the transverse hypersurface $\Sigma$ (in red). The intersection $\Sigma\cap Z$ is a plane parameterized by $\{\Upsilon= 0\}$. On the right, a view inside the neighborhood $U=J_\mu \cap Z$ (grey disc). It is foliated by the $G_\mu$-orbits, which are the integral curves of the Hamiltonian vector field $X_{\Phi}$ related to the first-class constraint $\Phi$ (green dashed circles). Intersecting with $\Sigma$ we have the slice $N=\Sigma\cap U = \{\Phi=0,\Upsilon=0\}$ (red segment) through $x_0$, meeting each nearby orbit. Thus the pair of constraint functions $(\Phi,\Upsilon)$ parameterizes the directions along and transverse to the characteristic foliation and defines the second-class constraint.
• Figure 6.1: Double spherical pendulum in two different configurations. On the left, the spinning relative equilibrium on a nonzero momentum level ($\mu \ne 0$), with constant angular velocity $\Omega$ and vertical angular momentum $J=\mu e_ 3$. On the right, the static equilibrium with zero angular momentum ($\mu = 0$), whose image in the reduced space $M_0$ is a singular point corresponding to full $S^1$ isotropy.
• Figure 6.2: Representation of the spinning relative equilibria corresponding to cases $(3)$ and $(4)$, respectively. In case $(3)$, $q_1$ is horizontal; the first mass rotates about $e_3$, whereas the second one lines on the $e_3$-axis and is fixed in space. In case $(4)$, $q_2$ is horizontal and both masses rotate about $e_3$. Both configurations have trivial isotropy and occur on nonzero momentum levels.

Theorems & Definitions (28)

• Definition 2.1
• Lemma 2.2: dirac1958generalized, see also marsden2013introduction
• Remark 2.3
• Definition 2.4
• Definition 2.5
• Remark 3.1
• Definition 3.3
• Remark 3.4
• Theorem A
• Theorem B