Invariant Imbedding Relation for the Principles Of Invariance

SIO Reference 57-45. The principle of invariant imbedding. recently stated by Bellmr and Kalaba, is a rule of action which may be followed in the mathematical formulation of any of a wide class of problems concerned with transfer phenomena in general media. The statement of the principle generalizes the methodology originally developed by Ambarzumian and extended by Chandrasekhar in their studies of the transfer of radiation through scattering and absorbing media. In this note we exhibit an explicit analytic embodiment of the principle for radiative transfer and neutron transport processes. This symbolic statement of the principle - which we shall call the invariant imbedding relation - yields in particular the general symmetric forms of the principles of invariance for these processes. The semi-group features which generally are associated with an invariantly imbedded process are also implicit in the invariant imbedding relation. The present discussion will be limited to the steady state case on a class of inhomogeneous one-parameter carrier spaces (e.g., plane-parallel, cylindrical, spherical, and general one-parameter space filling families of surfaces)