# American Institute of Mathematical Sciences

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

## Lagrangian dynamics by nonlocal constants of motion

 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.

Citation: Gianluca Gorni, Gaetano Zampieri. Lagrangian dynamics by nonlocal constants of motion. Discrete and 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. [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.

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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.
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)
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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