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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References

[1]	F. T. Arecchi and R. Meucci, Chaos in Lasers, Scholarpedia, 2008. doi: 10.4249/scholarpedia.7066.
[2]	F. Calogero, Solutions of the one dimensional $n$-body problems with quadratic and/or inversely quadratic pair potentials, J. Mathematical Phys., 12 (1971), 419-436. doi: 10.1063/1.1665604.
[3]	G. Gorni and G. Zampieri, Revisiting Noether's theorem on constants of motion, J. Nonlinear Math. Phys., 21 (2014), 43-73. doi: 10.1080/14029251.2014.894720.
[4]	G. Gorni and G. Zampieri, Nonlocal variational constants of motion in dissipative dynamics, Differential Integral Equations, 30 (2017), 631-640.
[5]	G. Gorni and G. Zampieri, Nonstandard separation of variables for the Maxwell-Bloch conservative system, São Paulo J. Math. Sci., 12 (2018), 146-169. doi: 10.1007/s40863-017-0079-3.
[6]	G. Gorni and G. Zampieri, Nonlocal and nonvariational extensions of Killing- type equations, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 675-689. doi: 10.3934/dcdss.2018042.
[7]	G. Gorni, S. Residori and G. Zampieri, A quasi separable dissipative Maxwell- Bloch system for laser dynamics, Qual. Theory Dyn. Syst., 18 (2019), 371-381. doi: 10.1007/s12346-018-0290-3.
[8]	P. G. L. Leach, Lie symmetries and Noether symmetries, Appl. Anal. Discrete Math., 6 (2012), 238-246. doi: 10.2298/AADM120625015L.
[9]	R. Leone and T. Gourieux, Classical Noether theory with application to the linearly damped particle, European Journal of Physics, 36 (2015), 20 pp. doi: 10.1088/0143-0807/36/6/065022.