Introduction to the Geometry of Classical Dynamics

Preface The aim of this paper is to lead (in most elementary terms) an undergraduate student of Mathematics or Physics from the historical Newtonian-d’Alembertian dynamics up to the border with the modern (geometrical ) Lagrangian-Hamiltonian dynamics, without making any use of the traditional (analytical ) formulation of the latter. Our expository method will in principle adopt a rigorously coordinate-free language, apt to gain – from the very historical formulation – the ‘consciousness’ (at an early stage) of the geometric structures that are ‘intrinsic’ to the very nature of classical dynamics. The coordinate formalism will be confined to the ancillary role of providing simple proofs for some geometric results (which would otherwise require more advanced geometry), as well as re-obtaining the local analytical formulation of the theory from the global geometrical one. The main conceptual tool of our approach will be the simple and general notion of differential equation in implicit form, which, treating an equation just as a subset extracted from the tangent bundle of some manifold through a geometric or algebraic property, will directly allow us to capture the structural core underlying the evolution law of classical dynamics.