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  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 & 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. GorniS. 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. GorniS. 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)
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)
[1]

Helmut Abels, Johannes Kampmann. Existence of weak solutions for a sharp interface model for phase separation on biological membranes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 331-351. doi: 10.3934/dcdss.2020325

[2]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273

[3]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[4]

Shao-Xia Qiao, Li-Jun Du. Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity. Electronic Research Archive, , () : -. doi: 10.3934/era.2020116

[5]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[6]

Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326

[7]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[8]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383

[9]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

[10]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[11]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267

[12]

Anna Abbatiello, Eduard Feireisl, Antoní Novotný. Generalized solutions to models of compressible viscous fluids. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 1-28. doi: 10.3934/dcds.2020345

[13]

Peter H. van der Kamp, D. I. McLaren, G. R. W. Quispel. Homogeneous darboux polynomials and generalising integrable ODE systems. Journal of Computational Dynamics, 2021, 8 (1) : 1-8. doi: 10.3934/jcd.2021001

[14]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

[15]

Monia Capanna, Jean C. Nakasato, Marcone C. Pereira, Julio D. Rossi. Homogenization for nonlocal problems with smooth kernels. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020385

[16]

Federico Rodriguez Hertz, Zhiren Wang. On $ \epsilon $-escaping trajectories in homogeneous spaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 329-357. doi: 10.3934/dcds.2020365

[17]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458

[18]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048

[19]

Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, 2021, 20 (1) : 405-425. doi: 10.3934/cpaa.2020274

[20]

Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (95)
  • HTML views (273)
  • Cited by (0)

Other articles
by authors

[Back to Top]