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A microcanonical kinetic theory of reactions based upon the structure within phase space is developed. It is shown that the dynamics of reaction across an energetic barrier is mediated by invariant manifolds embedded in phase space that have the geometry of simple cylinders. The ideas are developed by considering molecular systems modeled by two vibrational degrees of freedom, a reaction coordinate and a "bath" coordinate. The kinetic theory is constructed by focusing on the dynamics between n mapping planes (Un-map") and the "reactive island" (RI) structure within them. We discuss how the structure of the conformer population decay in isomerization reactions can be obtained from the RI kinetic model. Formal solutions of the kinetic equations are discussed with specific attention given towards the calculation of the isomerization reaction rate. The formal theory is developed in Paper I of this series. Numerical considerations and applications to the reaction dynamics of model molecular systems with two degrees of freedom will be given in Paper II and extension of the theory and applications to multidimensional systems will be given in Paper III. .

The Hamiltonian flow of a system of two degrees of freedom, capable of undergoing geometrical isomerization, is studied as a function of the coupling between the reactive and unreactive degrees of freedom. The reaction dynamics are analyzed from the perspective of the KAM theorem and the transition to chaos. Conditions are found for the validity of linear rate laws and RRKM theory in isolated molecules.