Variational aspects of analytical mechanics

Abstract

In this paper we set out to revisit some basic variational aspects of Analytical Mechanics in one independent variable, in a way that would be most appealing to those who are being introduced to Mechanics while having some background in elementary Functional Analysis. A slightly longer version [7], written in Italian, is going to appear in a supplementary volume to the reedition of the classical 1923 Lectures in Rational Mechanics by Levi-Civita and Amaldi [9]. We are going to start with

Hamilton’s principle, that we will state for systems that admit a Lagrangian function, that is just assumed to have continuous first partial derivatives. This minimal regularity is by no means new, even though it is unusual in Mechanics textbooks. We think that it leads to a more general and elegant theory, that may be useful also outside of Mechanics. We will then proceed to Emmy Noether’s theorem, a result whose first formulation goes back to 1918 in the context of PDE, and is still today an active source of inspiration for physical theories. We will only treat the one independent variable, ODE case.