Nonlinear mechanics is now a mature science. It has successfully been applied to a diversity of fields that show complex behaviors. As far as Chemistry is concerned nonlinear mechanics has helped us to comprehend concepts such as Transition State, Activated Complex, and Reaction Pathways, which are the cornerstones in statistical theories of chemical reactions. For first time these concepts have been put on a rigorous mathematical ground and extended to polyatomic molecules with multivariable Potential Energy Surfaces and complicated topologies. The application of the theory of nonlinear mechanics to molecular dynamics had as a consequence to elevate the stage on which chemical reactions are performed from the configuration manifold to its cotangent bundle (phase space). The latter revealed that time invariant phase space structures, such as Periodic Orbits, Tori, Normally Hyperbolic Invariant Manifolds (NHIMs) and their (un)stable manifolds are the mathematical objects that should be associated with the transition state and reaction pathways. It has been demonstrated that bifurcations of periodic orbits, such as the center-saddle, define reaction paths in phase space, which can not be predicted from the landscape of the potential functions.
I shall present classical and exact quantum dynamical calculations for realistic cases of small triatomic molecules spectroscopically studied, and model systems for large biological molecules, which show that nonlinear mechanics is ineluctable for understanding how chemical bonds break/form in elementary chemical reactions and the necessity of moving from structural chemistry to phase space chemistry.