Spin Calogero models obtained from dynamical r-matrices and geodesic motion

We study classical integrable systems based on the Alekseev-Meinrenken dynamical r-matrices corresponding to automorphisms of self-dual Lie algebras, ${\cal G}$. We prove that these r-matrices are uniquely characterized by a non-degeneracy property and apply a construction due to Li and Xu to associate spin Calogero type models with them. The equation of motion of any model of this type is found to be a projection of the natural geodesic equation on a Lie group $G$ with Lie algebra ${\cal G}$, and its phase space is interpreted as a Hamiltonian reduction of an open submanifold of the cotangent bundle $T^*G$, using the symmetry arising from the adjoint action of $G$ twisted by the underlying automorphism. This shows the integrability of the resulting systems and gives an algorithm to solve them. As illustrative examples we present new models built on the involutive diagram automorphisms of the real split and compact simple Lie algebras, and also explain that many further examples fit in the dynamical r-matrix framework.