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.