Characterization of Vector Fields on the Frölicher Standard n-Simplex, Imbedded in R ⁿ , and Hamiltonian Formalism on Differential Spaces

Abstract

This paper is an attempt to the symplectization of smooth unusual, but standard imbedded subspaces of R ⁿ like a n-simplex, for the purpose of modelling Hamiltonian and Lagrangian systems thereon. We show that these subspaces are obtained by the process of smoothly gluing portions of R ⁿ , all considered as Frölicher spaces. On working with some of them, we characterize smooth vector fields that have a flow. Then, we show that both the modern and classical mechanical systems can be written in a more larger category than that of Frölicher spaces, which is the category of differential spaces.