Manifolds and Catastrophes for Physical Systems

Abstract

New geometrical considerations are requested recently form physics. This arises after the finding of gravitational waves, demonstrating the length contraction/expansion as space-time riffles. In earlier publications, the author has published such geometries, dynamics, and shapes for physical systems, adding different kinds of mathematics. In this article, the focus is on manifolds which locally are like a real or complex Euclidean 4-dimensional (Hilbert) space H4. The influence of potential functions from catastrophe theory is investigated. They are a r-parameter family of smooth functions Rn→R for natural numbers n and all r ≤ 5. There are seven catastrophes in Thom’s theory which are structurally stable.[10] Besides, the catastrophe manifolds other manifolds arise, the non-critical case without polar singularities and the non-degenerate critical Morse functions case with a diagonal matrix for the metric diag (Ij−In−i), I the identity matrix, j≤n. Duals of catastrophes are not mentioned. Complex numbers, quaternion, and octonion vectorial extensions are added to H4. They are not used for the subspace lattice L of H4 which remains 4-dimensional. L is the union of its Boolean blocks for sets of commuting projection operators H4 → Upe which split H4 = U + U┴, U┴ the orthogonal complement of the closed subspace U. The block structure requests several catastrophes which are discussed in the first section. Applications are then added for bifurcations, Gleason measuring spin-like frames and for systems such as particles, quasi-articles, or effects on states of systems.