Scaling Symmetries and Conformal Relative Equilibria on Poisson Manifolds, with Applications to Lie-Poisson Systems

We investigate conformal relative equilibria for Hamiltonian systems on exact Poisson manifolds equipped with scaling symmetries. By introducing conformally Poisson actions and conformal momentum maps, we characterize these equilibria through an augmented Hamiltonian formulation; in the nondegenerate case, this recovers the conditions recently developed for the exact symplectic case. Specializing to the Lie-Poisson manifold $\mathfrak g^*$ associated with an arbitrary finite-dimensional real Lie algebra $\mathfrak g$, we show that its natural scaling action makes $\mathfrak g^*$ an exact Poisson manifold. We then establish a purely algebraic existence criterion: there exists a homogeneous Hamiltonian on $\mathfrak g^*$ admitting a nontrivial conformal relative equilibrium if and only if $\mathfrak g$ contains an element whose adjoint operator has a nonzero real eigenvalue. Applying this criterion to the Bianchi classification, we identify exactly the real three-dimensional Lie algebras for which there exists a homogeneous Hamiltonian admitting a nontrivial conformal relative equilibrium. As a prominent application, we show that nontrivial conformal relative equilibria exist for a rigid body on $\mathfrak{so}(2,1)^*$, whereas the classical free rigid body on $\mathfrak{so}(3)^*$ admits no nontrivial conformal relative equilibria.