Lagrangian dynamics by nonlocal constants of motion

Gianluca Gorni; Gaetano Zampieri

doi:10.3934/dcdss.2020216

Article Contents

2020, Volume 13, Issue 10: 2751-2759. Doi: 10.3934/dcdss.2020216

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Lagrangian dynamics by nonlocal constants of motion

Gianluca Gorni^1, and
Gaetano Zampieri^2, ,

1.
Università di Udine, Dipartimento di Scienze Matematiche, Informatiche e Fisiche, via delle Scienze 208, 33100 Udine, Italy
2.
Università di Verona, Dipartimento di Informatica, strada Le Grazie 15, 37134 Verona, Italy

^* Corresponding author: Gaetano Zampieri
^* Corresponding author: Gaetano Zampieri

Received: January 2019

Revised: June 2019

Early access: December 2019

Published: July 2020

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Abstract

A simple general theorem is used as a tool that generates nonlocal constants of motion for Lagrangian systems. We review some cases where the constants that we find are useful in the study of the systems: the homogeneous potentials of degree $ -2 $, the mechanical systems with viscous fluid resistance and the conservative and dissipative Maxwell-Bloch equations of laser dynamics. We also prove a new result on explosion in the past for mechanical system with hydraulic (quadratic) fluid resistance and bounded potential.

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Mathematics Subject Classification: 37J15, 70H33.

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Figure 1. Generic (periodic) and homoclinic orbits of $ (\dot q_3, \ddot q_3) $ in the conservative case of the Maxwell-Bloch equations (Subsec. 6.1); the dashed lines and the dots are not visited by solutions, but they are level sets or stationary points of the potential function associated with equation (35)

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Figure 2. Projection of forward orbits on the $ q_1, q_2 $ plane in two dissipative cases with $ c = 2a $ of the Maxwell-Bloch equations (Subsec. 6.2), computed numerically. On the left with $ g^2k\le ab $ the solution goes to the origin; on the right with $ g^2k> ab $ the orbit converges to a point on the (dashed) circle with radius $ r_\infty $, as in equation (44)

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