Numerical evidence for the approximation of dissipative systems by gyroscopically coupled oscillator chains

Inspired by the equations of motion of the gyroscope, where a Lagrangian and conservative system may appear to mimic a dissipative one when focusing on a single degree of freedom for finite time intervals, we introduce a gyroscopic-type coupling between harmonic oscillators. The aim is to propose a tentative scheme of solution for the problem of finding a higher-dimensional Lagrangian system approximating a lower-dimensional dissipative one. Specifically, we consider a certain family of Lagrangian systems, for which the time evolution of the first Lagrangian parameter is conjectured to be a good approximation for the evolution of a one-dimensional linear dissipative system in finite time intervals, up to a fixed precision. The behavior of the selected family of gyroscopic couplings is compared with a given dissipative system, properly optimizing a family of parameters according to a described scheme. Numerical calculations are reported, suggesting the validity of the proposed conjecture

Lagrange formal calculus as applied to Lagrange mechanics: an exercise in anachronism

When Lagrange wrote his masterpiece Mecanique Analytique, the foundations of analysis were not completely understood: to erect the great building of Analytical Mechanics upon solid foundations, the Piedmontese mathematician tried to lay the foundations of differential calculus in a purely algebraic way, using power series instead of functions, regardless about convergence and uniqueness issues. While this foundation was unsatisfactory as shown by Cauchy some decades later, it can shed light on how Lagrange considered the analytical objects (curves, energies, etc.) he dealt with in Mechanics. In this paper, we review these Lagrangian foundations of analysis, and we try to adopt its obvious modern counterpart, i.e., formal power series, to express some results in Analytical Mechanics related to Helmholtz conditions and Rayleigh description of dissipation. By means of purely algebraic manipulations, we will easily recover results otherwise proved by means of modern analysis

Approximation of dissipative systems by elastic chains: numerical evidence

An old and debated problem in Mechanics concerns the capacity of finite dimensional Lagrangian systems to describe dissipation phenomena. It is true that Helmholtz conditions determine not-always verifiable conditions establishing when a system of n second-order ordinary differential equations in normal form (nODEs) be the Lagrange equations deriving from an nth dimensional Lagrangian. However, it is also true that one could conjecture that, given nODEs it is possible to find a (n+k)th dimensional Lagrangian such that the evolution of suitably chosen n Lagrangian parameters allows for the approximation of the solutions of the nODEs. In fact, while it is well known that the ordinary differential equations (ODEs) usually introduced for describing some dissipation phenomena do not verify Helmholtz conditions, in this paper, we give some preliminary evidence for a positive answer to the conjecture that a dissipative system having n degrees of freedom (DOFs) can be approximated, in a finite time interval and in a suitable norm, by an extended Lagrangian system, having a greater number of DOFs. The theoretical foundation necessary to formulate such a conjecture is here laid and three different examples of extended Lagrangians are shown. Finally, we give some computational results, which encourage to deepen the study of the theoretical aspects of the problem

Il fascino della matematica e delle sue applicazioni

Il fascino della matematica e delle sue applicazioni e` un progetto del Dipartimento di Scienze di Base e Applicate per l'Ingegneria finanziato nell'ambito de bando terza missione per l'anno 2020 di Sapienza Universita` di Roma. L'iniziativa ha avuto come scopo quello di diffondere non solo gli aspetti piu` intriganti della matematica, ma anche le sue ricadute sulle scienze applicate, attraverso una serie di seminari, tenuti da docenti universitari e docenti di scuola superiore, rivolti principalmente a un pubblico di studenti della scuola superiore. Il testo raccoglie il contributo di alcuni degli oratori coinvoti nei seminari