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October  2020, 13(10): 2751-2759. doi: 10.3934/dcdss.2020216

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

Received  January 2019 Revised  June 2019 Published  October 2020 Early access  December 2019

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.

Keywords: Nonlocal constants of motion, homogeneous potentials, viscous fluid resistance, hydraulic fluid resistance, Maxwell-Bloch equations, global existence, explosion in the past, nonstandard separation of variables.
Mathematics Subject Classification: 37J15, 70H33.
Citation: Gianluca Gorni, Gaetano Zampieri. Lagrangian dynamics by nonlocal constants of motion. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2751-2759. doi: 10.3934/dcdss.2020216
References:
[1]

F. T. Arecchi and R. Meucci, Chaos in Lasers, Scholarpedia, 2008. doi: 10.4249/scholarpedia.7066.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

G. Gorni and G. Zampieri, Nonlocal variational constants of motion in dissipative dynamics, Differential Integral Equations, 30 (2017), 631-640.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

P. G. L. Leach, Lie symmetries and Noether symmetries, Appl. Anal. Discrete Math., 6 (2012), 238-246.  doi: 10.2298/AADM120625015L.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

F. T. Arecchi and R. Meucci, Chaos in Lasers, Scholarpedia, 2008. doi: 10.4249/scholarpedia.7066.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

G. Gorni and G. Zampieri, Nonlocal variational constants of motion in dissipative dynamics, Differential Integral Equations, 30 (2017), 631-640.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

P. G. L. Leach, Lie symmetries and Noether symmetries, Appl. Anal. Discrete Math., 6 (2012), 238-246.  doi: 10.2298/AADM120625015L.  Google Scholar

[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.  Google Scholar

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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