Infinitesimal calculus of variations (Q1805702)

The author seeks to develop an infinitesimal calculus of variations in the context of synthetic differential geometry (SDG). He assumes that the reader is familiar with the basic underpinnings and philosophy of SDG as developed in the recent text by \textit{R. Lavendhomme} [`Basic concepts of synthetic differential geometry' (Kluwer, Math. Sci. 13, Kluwer Texts Dordrecht) (1996; Zbl 0866.58001)] although some of the needed definitions are summarized in the first section. The basic aim of the paper is to develop a version of Lagrange's equation in this setting, and then this is used to derive generalized conservation laws of momentum and energy. The basic setting for these derivations is on a microlinear space \(M\) with a symmetric connection with an associated real-line valued function \(L\) (Lagrangian) on the tangent bundle of \(M\).