Generalized double bracket vector fields

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.