Collective modes of interacting many-body systems can be related to motion on classically invariant manifolds.We introduce suitable coordinate systems.These coordinates are Cartesian in position and momentum space.They are collective since several components vanish for motion on the invariant manifold.We make a connection to Zickendraht's collective coordinates and also obtain shear modes.The importance of collective configurations depends on the stability of the manifold.We present an example of quantum collective motion on the manifold.