The general differential-geometric structure of multidimensional Delsarte transmutation operators in parametric functional spaces and their applications in soliton theory. Part 2

The structure properties of multidimensional Delsarte transmutation operators in parametirc functional spaces are studied by means of differential-geometric tools. It is shown that kernels of the corresponding integral operator expressions depend on the topological structure of related homological cycles in the coordinate space. As a natural realization of the construction presented we build pairs of Lax type commutive differential operator expressions related via a Darboux-Backlund transformation having a lot of applications in solition theory. Some results are also sketched concerning theory of Delsarte transmutation operators for affine polynomial pencils of multidimensional differential operators.Comment: 10 page

The Electromagnetic Lorentz Problem and the Hamiltonian Structure Analysis of the Maxwell-Yang-Mills Type Dynamical Systems within the Reduction Method

Based on analysis of reduced geometric structures on fi bered manifolds, invariant under action of an abelian functional symmetry group, we construct the symplectic structures associated with connection forms on the related principal fi ber bundles with abelian functional structure groups. The Marsden-Weinstein reduction procedure is applied to the Maxwell and Yang-Mills type dynamical systems. The geometric properties of Lorentz type constraints, describing the electromagnetic fi eld properties in vacuum and related with the well known Dirac-Fock-Podolsky problem, are discussed

The differential-algebraic and bi-Hamiltonian integrability analysis of the Riemann type hierarchy revisited

A differential-algebraic approach to studying the Lax type integrability of the generalized Riemann type hydrodynamic hierarchy is revisited, its new Lax type representation and Poisson structures constructed in exact form. The related bi-Hamiltonian integrability and compatible Poissonian structures of the generalized Riemann type hierarchy are also discussed.Comment: 18 page

The Electromagnetic Lorentz Condition Problem and Symplectic Properties of Maxwell and Yang-Mills Type Dynamical Systems

Symplectic structures associated to 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 of dynamical systems of Maxwell and Yang-Mills type. 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. 1

The Relativistic Electrodynamics Least Action Principles Revisited: New Charged Point Particle and Hadronic String Models Analysis

The classical relativistic least action principle is revisited from the vacuum field theory approach. New physically motivated versions of relativistic Lorentz type forces are derived, a new relativistic hadronic string model is proposed and analyzed in detail.Comment: n/

The Vacuum Structure, Special Relativity and Quantum Mechanics Revisited: a Field Theory No-Geometry Approach within the Lagrangian and Hamiltonian Formalisms. Part 2

The main fundamental principles characterizing the vacuum field structure are formulated and the modeling of the related vacuum medium and charged point particle dynamics by means of devised field theoretic tools are analyzed. The work is devoted to studying the vacuum structure, special relativity, electrodynamics of interacting charged point particles and quantum mechanics, and is a continuation of \cite{BPT,BRT1}. Based on the vacuum field theory no-geometry approach, the Lagrangian and Hamiltonian reformulation of some alternative classical electrodynamics models is devised. The Dirac type quantization procedure, based on the canonical Hamiltonian formulation, is developed for some alternative electrodynamics models. Within an approach developed a possibility of the combined description both of electrodynamics and gravity is analyzed.Comment: 11 page