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Abstract. 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.

The authors dedicate this article to their friend and teacher academician Prof. Anatoliy M. Samoilenko on the occasion of his 70th Birthday with great compliments and gratitude to his brilliant talent and impressive impact to modern theory of nonlinear dynamical systems of mathematical physics and nonlinear analysis.

Relying on theoretical developments exploiting quasispin and the pseudo-orthogonal group in the Hubbard model of cyclic polyenes, the general expressions for generating polynomials, providing the dimensional information for relevant irreducible representations, were derived (M. D. Could, J. Paldus, and J. Ciiek, Int. J. Quantum Chem., in press). These generating polynomials result from qdimensional formulas through rather tedious algebraic manipulations involving ratios of polynomials with fractional powers. It is shown that these expressions may be efficiently handled using the symbolic manipulation language MAPLE and the dimensional information for an arbitrary spin, isospin, and quasimomentum obtained. Exploitation of symbolic computation for other group theoretical problems that are relevant in quantum chemical calculations and their relationship with Gaussian polynomial based combinatorial approaches is also briefly addressed and various possible applications outlined. 0 1994 John Wiley & Sons, Inc.

UDC 517.9The backgrounds of quantum mathematics, a new discipline in mathematical physics, are discussed and analyzed from both historical and analytical points of view. The magic properties of the second quantization method, invented by Fock in 1934, are demonstrated, and an impressive application to the theory of nonlinear dynamical systems is considered.