Hamiltonian evolution of curves in classical affine geometries

In this paper we study geometric Poisson brackets and we show that, if M = ( G ⋉ R n ) / G endowed with an affine geometry (in the Klein sense), and if G is a classical Lie group, then the geometric Poisson bracket for parametrized curves is a trivial extension of the one for unparametrized curves, except for the case G = GL ( n , R ) . This trivial extension does not exist in other nonaffine cases (projective, conformal, etc).