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

Alberto Maria Bersani, Paolo Caressa and Alessandro Ciallella

Vol. 10 (2022), No. 3, 265–278
Abstract

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.

Keywords
Lagrangian formalism, approximation of dissipative systems, conservative systems, gyroscopes
Mathematical Subject Classification
Primary: 70-08, 70-10, 70H03, 70H05, 70H20
Milestones
Received: 20 February 2022
Revised: 3 June 2022
Accepted: 18 August 2022
Published: 4 December 2022

Communicated by Francesco dell'Isola
Authors
Alberto Maria Bersani
Dipartimento di Ingegneria Meccanica e Aerospaziale
Sapienza Università di Roma
Rome
Italy
Gruppo Nazionale di Fisica Matematica
Italy
International Research Center for the Mathematics and Mechanics of Complex Systems
University of L’Aquila
Italy
Paolo Caressa
Gestore dei Servizi Energetici
Rome
Italy
Alessandro Ciallella
Department of Civil, Construction-Architectural and Environmental Engineering
University of L’Aquila
Italy
International Research Center for the Mathematics and Mechanics of Complex Systems
University of L’Aquila
Italy