  We show that a large class of physical theories which has been under
intensive investigation recently, share the same geometric features in their
Hamiltonian formulation. These dynamical systems range from harmonic
oscillations to WZW-like models and to the KdV dynamics on $Diff_oS^1$. To the
same class belong also the Hamiltonian systems on groups of maps.
  The common feature of these models are the 'chiral' equations of motion
allowing for so-called chiral decomposition of the phase space.
