Classical mechanics coupled identical oscillators

The classical mechanical problem of two coupled identical harmonic oscillators is described by the classical mechanical Hamiltonian... 

The phase space of a coupled, two-identical-anharmonic oscillator system is four-dimensional. Conservation of energy and polyad number reduces the number of independent variables from four to two. At specified values of E and N = vr + vl = vs+ v0 (in classical mechanics, N need no longer be restricted to integer values nor E to eigenenergies), accessible phase space divides into several distinct regions of regular, qualitatively describable motions and (for more general dynamical systems) chaotic, indescribable motions. Systematic variation of E and N reveals bifurcations in the number of forms of these describable motions. Examination of the classical mechanical form of the polyad Heff often reveals the locations and causes of such bifurcations. 

In order to illustrate the relationship between the quantum mechanical Heff and the classical mechanical Ti derived from it by applying Heisenberg s (1925) version of the Correspondence Priniciple, we return to the problem of two coupled identical anharmonic oscillators (see Section 9.4.12 and Xiao and Kellman, 1989). The quantum mechanical HlqCAL is,... 

The phase space structures for two identical coupled anharmonic oscillators are relatively simple because the trajectories lie on the surface of a 2-dimensional manifold in a 4-dimensional phase space. The phase space of two identical 2-dimensional isotropic benders is 8-dimensional, the qualitative forms of the classifying trajectories are far more complicated, and there is a much wider range of possibilities for qualitative changes in the intramolecular dynamics. The classical mechanical polyad 7feff conveys unique insights into the dynamics encoded in the spectrum as represented by the Heff fit model. 

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