Generalized double bracket vector fields
Petre Birtea, Zohreh Ravanpak, Cornelia Vizman
TL;DR
This work extends the classical double bracket construction from compact semisimple Lie algebras to general Poisson manifolds endowed with a (pseudo-)Riemannian metric by introducing a symmetric contravariant 2-tensor ${\mathcal M}$ (the metriplectic tensor). The generalized double bracket (GDB) vector field $\partial_{\mathcal M}G$ is tangent to symplectic leaves and, on good leaves, acts as a gradient with respect to a newly defined DB metric, unifying conservative and dissipative dynamics without requiring metric-Poisson compatibility. A key contribution is the gradient interpretation on leaves, the notion of good symplectic leaves, and the demonstration that compact semisimple cases recover the negative normal metric. Practically, the framework provides a dissipation mechanism to stabilize already-stable equilibria within Hamiltonian systems, as shown in a resonant two-oscillator example, with potential broad applicability in geometric mechanics and dissipative system design.
Abstract
We generalize double bracket vector fields, originally defined on semisimple Lie algebras, to Poisson manifolds equipped with a pseudo-Riemannian metric by utilizing a symmetric contravariant 2-tensor field. We extend the normal metric on an adjoint orbit of a compact semisimple Lie algebra to ensure that these vector fields become gradient vector fields on each symplectic leaf. Furthermore, we apply this construction to enhance the equilibria of Hamiltonian systems, specifically addressing the challenge of asymptotically stabilizing points that are already stable, through dissipation terms derived from generalized double bracket vector fields.