Synthetic Hamiltonian mechanics (Q678648)

The paper deals with some infinitesimal aspects of Hamiltonian mechanics from the standpoint of synthetic differential geometry. The latter involves the machinery of ``infinitesimals'' in a way distinct from the non-standard analysis. The reader is referred to the monographs [\textit{A. Kock}, Synthetic differential geometry, Cambridge University Press (1981; Zbl 0466.51008), \textit{R. Lavendhomme}, `Leçons de géométrie différentielle synthétique naïve,' (1987; Zbl 0688.18006), \textit{I. Moerdijk} and \textit{G. E. Reyes}, `Models for smooth infinitesimal analysis', Springer (1991; Zbl 0715.18001), etc.] for its description. Here fundamental results and constructions connected with Hamiltonian vector fields, Poisson brackets, momentum mappings, Lie derivatives, etc. are discussed. The paper consists of two sections (subdivided into subsections). The first one is concerned with the synthetic differential geometry and the second section is devoted to synthetic Hamiltonian mechanics itself.