We recently developed a new method to extract a many-body phase-space dividing surface, across which the transmission coefficient for the classical reaction path is unity. The example of isomerization of a 6-atom Lennard-Jones cluster showed that the action associated with the reaction coordinate is an approximate invariant of motion through the saddle regions, even at moderately high energies, at which most or all the other modes are chaotic [J. Chem. Phys. 105, 10838 (1999); Phys. Chem. Chem. Phys. 1, 1387 (1999)]. In the present article, we propose a new algorithm to analyze local invariances about the transition state of N-particle Hamiltonian systems. The approximate invariants of motion associated with a reaction coordinate in phase space densely distribute in the sea of chaotic modes in the region of the transition state. Using projections of distributions in only two principal coordinates, one can grasp and visualize the stable and unstable invariant manifolds to and from a hyperbolic point of a many-body nonlinear system, like those of the one-dimensional, integrable pendulum. This, in turn, reveals a new type of phase space bottleneck in the region of a transition state that emerges as the total energy increases, which may trap a reacting system in that region.