| 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 |
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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F. T. Arecchi and R. Meucci, Chaos in Lasers, Scholarpedia, 2008.
doi: 10.4249/scholarpedia.7066. |
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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. |
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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. |
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G. Gorni and G. Zampieri,
Nonlocal variational constants of motion in dissipative dynamics, Differential Integral Equations, 30 (2017), 631-640.
|
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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. |
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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
| [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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