This brief presents numerical methods for describing and calculating invariant phase space
structures as well as solving the classical and quantum equations of motion for polyatomic
molecules. Examples covered include simple model systems to realistic cases of molecules
spectroscopically studied. Vibrationally excited and reacting molecules are nonlinear dynamical
systems and thus nonlinear mechanics is the proper theory to elucidate molecular dynamics by
investigating invariant structures in phase space. Intramolecular energy transfer and the
breaking and forming of a chemical bond have now found a rigorous explanation by studying phase
space structures.
