2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/58401A comparative analysis of two different versions of the Legendre transformation is presented. We provide an almost complete although somewhat superficial review of the geometric background for analytical mechanics. Complete coordinate characterizations of all structures are provided. Intrinsic constructions of most of the objects are given. Examples of applications to a number of physical systems is given50 pages, LamsTeX. To appear in Acta Physica PolonicaMathematical PhysicsDifferential Geometrytext