We formulate a coarse-grained mean-field approach to study the dynamics of the flexible ring polymer in any given obstacle (gel or melt) environment. The similarity of the static structure of the ring polymer with that of the ideal randomly branched polymer is exploited in formulating the dynamical model using aspects of the pom-pom model for branched polymers. The topological constraints are handled via the tube model framework. Based on our formulation we obtain expressions for diffusion coefficient D, relaxation times tau, and dynamic structure factor g(k,t). Further, based on the framework we develop a molecular theory of linear viscoelasticity for ring polymers in a given obstacle environment and derive the expression for the relaxation modulus G(t). The predictions of the theoretical model are in agreement with previously proposed scaling arguments and in qualitative agreement with the available experimental results for the melt of rings.

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