Symmetry in Nonlinear Mathematical Physics - 2009

Anatoliy Prykarpatsky (Drohobych Ivan Franko State Pedagogical University, Ukraine & AGH University of Science and Technology, Krakow, Poland)

On the symmetry structure of the Dirac-Fock-Podolsky electrodynamic problem

Symplectic structures associated with connection forms on certain types of principal fiber bundles are constructed via analysis of reduced geometric structures on fibered manifolds invariant under naturally related symmetry groups. This approach is then applied to nonstandard Hamiltonian analysis of dynamical systems of Maxwell and Yang–Mills types. A symplectic reduction theory of the classical Maxwell equations is formulated so as to naturally include the Lorentz condition (ensuring the existence of electromagnetic waves), thereby solving the well-known Dirac–Fock–Podolsky problem. Symplectically reduced Poissonian structures and the related classical minimal interaction principle for the Yang–Mills equations are also considered.