GREEN FORMULA AND VARIATIONAL-PRINCIPLES FOR COSMIC-RAY TRANSPORT WITH APPLICATION TO ROTATING AND SHEARING FLOWS

Green's theorem and Green's formula for the diffusive cosmic-ray trans port equation in relativistic flows are derived. Green's formula gives the solution of the transport equation in terms of the Green's functi on of the adjoint transport equation, and in terms of distributed sour ces throughout the region R of interest, plus terms involving the part icle intensity and streaming on the boundary. The adjoint transport eq uation describes the time-reversed particle transport. An Euler-Lagran ge variational principle is then obtained for both the mean scattering frame distribution function f and its adj oint f dagger. Variations o f the variational functional with respect to f dagger yield the transp ort equation, whereas variations of f yield the adjoint transport equa tion. The variational principle, when combined with Noether's theorem, yields the conservation law associated with Green's theorem. An inves tigation of the transport equation for steady, azimuthal, rotating flo ws suggests the introduction of a new independent variable H to replac e the comoving frame momentum variable p'. For the case of rigid rotat ing flows, H is conserved and is shown to be analogous to the Hamilton ian for a bead on a rigidly rotating wire. The variable H corresponds to a balance between the centrifugal force and the particle inertia in the rotating frame. The physical interpretation of H includes a discu ssion of nonrelativistic and special relativistic rotating flows as we ll as the cases of azimuthal, differentially rotating flows about Schw arzschild and Kerr black holes. Green's formula is then applied to the problem of the acceleration of ultra-high-energy cosmic rays by galac tic rotation. The model for galactic rotation assumes an angular veloc ity law OMEGA = OMEGA0(OMEGA0bar/omegaBAR), where omegaBAR denotes rad ial distance from the axis of rotation. Green's functions for the gala ctic rotation problem are used to investigate the spectrum of accelera ted particles arising from monoenergetic and truncated power-law sourc es. We conclude that it is possible to accelerate particles beyond the knee by galactic rotation but not in sufficient number to adequately explain the observed spectrum.