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. 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.

[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.

show all references

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. 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.

[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.

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]

Šárka Nečasová, Joerg Wolf. On the existence of global strong solutions to the equations modeling a motion of a rigid body around a viscous fluid. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1539-1562. doi: 10.3934/dcds.2016.36.1539

[2]

Kun Wang, Yangping Lin, Yinnian He. Asymptotic analysis of the equations of motion for viscoelastic oldroyd fluid. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 657-677. doi: 10.3934/dcds.2012.32.657

[3]

Lan Huang, Zhiying Sun, Xin-Guang Yang, Alain Miranville. Global behavior for the classical solution of compressible viscous micropolar fluid with cylinder symmetry. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1595-1620. doi: 10.3934/cpaa.2022033

[4]

Eduard Feireisl, Dalibor Pražák. A stabilizing effect of a high-frequency driving force on the motion of a viscous, compressible, and heat conducting fluid. Discrete and Continuous Dynamical Systems - S, 2009, 2 (1) : 95-111. doi: 10.3934/dcdss.2009.2.95

[5]

Bernard Ducomet, Šárka Nečasová. On the motion of rigid bodies in an incompressible or compressible viscous fluid under the action of gravitational forces. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1193-1213. doi: 10.3934/dcdss.2013.6.1193

[6]

Kun Wang, Yinnian He, Yueqiang Shang. Fully discrete finite element method for the viscoelastic fluid motion equations. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 665-684. doi: 10.3934/dcdsb.2010.13.665

[7]

Abelardo Duarte-Rodríguez, Lucas C. F. Ferreira, Élder J. Villamizar-Roa. Global existence for an attraction-repulsion chemotaxis fluid model with logistic source. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 423-447. doi: 10.3934/dcdsb.2018180

[8]

Huicheng Yin, Lin Zhang. The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅱ: 3D Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1063-1102. doi: 10.3934/dcds.2018045

[9]

Alessandro Bertuzzi, Antonio Fasano, Alberto Gandolfi, Carmela Sinisgalli. Interstitial Pressure And Fluid Motion In Tumor Cords. Mathematical Biosciences & Engineering, 2005, 2 (3) : 445-460. doi: 10.3934/mbe.2005.2.445

[10]

D. L. Denny. Existence of solutions to equations for the flow of an incompressible fluid with capillary effects. Communications on Pure and Applied Analysis, 2004, 3 (2) : 197-216. doi: 10.3934/cpaa.2004.3.197

[11]

Myeongju Chae, Kyungkeun Kang, Jihoon Lee. Existence of smooth solutions to coupled chemotaxis-fluid equations. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2271-2297. doi: 10.3934/dcds.2013.33.2271

[12]

Yue-Jun Peng, Shu Wang. Asymptotic expansions in two-fluid compressible Euler-Maxwell equations with small parameters. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 415-433. doi: 10.3934/dcds.2009.23.415

[13]

Min Li, Xueke Pu, Shu Wang. Quasineutral limit for the compressible two-fluid Euler–Maxwell equations for well-prepared initial data. Electronic Research Archive, 2020, 28 (2) : 879-895. doi: 10.3934/era.2020046

[14]

Timothy C. Reluga, Jan Medlock. Resistance mechanisms matter in SIR models. Mathematical Biosciences & Engineering, 2007, 4 (3) : 553-563. doi: 10.3934/mbe.2007.4.553

[15]

Shijin Ding, Boling Guo, Junyu Lin, Ming Zeng. Global existence of weak solutions for Landau-Lifshitz-Maxwell equations. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 867-890. doi: 10.3934/dcds.2007.17.867

[16]

Rémi Carles. Global existence results for nonlinear Schrödinger equations with quadratic potentials. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 385-398. doi: 10.3934/dcds.2005.13.385

[17]

Zaihui Gan, Boling Guo, Jian Zhang. Blowup and global existence of the nonlinear Schrödinger equations with multiple potentials. Communications on Pure and Applied Analysis, 2009, 8 (4) : 1303-1312. doi: 10.3934/cpaa.2009.8.1303

[18]

Feng Li, Yuxiang Li. Global existence of weak solution in a chemotaxis-fluid system with nonlinear diffusion and rotational flux. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5409-5436. doi: 10.3934/dcdsb.2019064

[19]

Seung-Yeal Ha, Bingkang Huang, Qinghua Xiao, Xiongtao Zhang. A global existence of classical solutions to the two-dimensional kinetic-fluid model for flocking with large initial data. Communications on Pure and Applied Analysis, 2020, 19 (2) : 835-882. doi: 10.3934/cpaa.2020039

[20]

Marco Di Francesco, Alexander Lorz, Peter A. Markowich. Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1437-1453. doi: 10.3934/dcds.2010.28.1437

2021 Impact Factor: 1.865

Metrics

  • PDF downloads (245)
  • HTML views (277)
  • Cited by (0)

Other articles
by authors

[Back to Top]