Rotational coordinates conjugate momenta

In analogy with the one-dimensional analysis, the Jj are defined over complete periods of the orbit in the (qj,Pj) plane, Jj = ptdq j. If one of the separation coordinates is cyclic, its conjugate momentum is constant. The corresponding orbit in the (qj,Pj) plane of phase space is a horizontal straight line, which may be considered as the limiting case of rotational periodicity, for which the cyclic qj always has a natural period of 2-k, and Jj = 2irpj for all cyclic variables. 

If we denote the coordinates corresponding to the translational and rotational motions of the molecule by Rj and its conjugate momentum by pg. 

Let us suppose that the liquid system is described by a MFPKE in N + 1 rigid bodies (the solute, or body 1 and N rotational solvent modes or bodies ), each characterized by inertia and friction tensors I and a set of Euler angles ft , and an angular momentum vector L (n = 1,..., N -I-1) plus K fields, each defined by a generalized mass tensor and friction tensor and a position vector and the conjugate linear momentum k = 1,..., K). The time evolution of the joint conditional probability x", L , P° 11, X, L, P, t) (where ft, X, etc. stand for the collection of Euler angles, field coordinates etc.) for the system is governed by the multivariate Fokker-Planck-Kramers equation... 

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