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August  2018, 11(4): 675-689. doi: 10.3934/dcdss.2018042

Nonlocal and nonvariational extensions of Killing-type equations

Università di Udine, Dipartimento di Scienze Matematiche, Informatiche e Fisiche, via delle Scienze 208, 33100 Udine, Italy

Università di Verona, Dipartimento di Informatica, strada Le Grazie 15, 37134 Verona, Italy

* Corresponding author: Gaetano Zampieri.

Received  November 2016 Revised  May 2017 Published  November 2017

The Killing-like equation and the inverse Noether theorem arise in connection with the search for first integrals of Lagrangian systems. We generalize the theory to include "nonlocal" constants of motion of the form $N_0+∈t N_1\, dt$, and also to nonvariational Lagrangian systems $\frac{d}{dt}\partial_{\dot q}L-\partial_qL=Q$. As examples we study nonlocal constants of motion for the Lane-Emden system, for the dissipative Maxwell-Bloch system and for the particle in a homogeneous potential.

Citation: Gianluca Gorni, Gaetano Zampieri. Nonlocal and nonvariational extensions of Killing-type equations. Discrete and Continuous Dynamical Systems - S, 2018, 11 (4) : 675-689. doi: 10.3934/dcdss.2018042
References:
[1]

F. T. Arecchi and R. Meucci, Chaos in lasers, Scholarpedia, 3 (2008), 7066. doi: 10.4249/scholarpedia.7066.

[2]

L. Y. Bahar and H. G. Kwatny, Extension of Noether's theorem to constrained non-conservative dynamical systems, Int. J. Non-Linear Mechanics, 22 (1987), 125-138.  doi: 10.1016/0020-7462(87)90015-1.

[3]

F. Calogero, Solutions of the one dimensional $n$-body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys, 12 (1971), 419-436.  doi: 10.1063/1.1665604.

[4]

E. CandottiC. Palmieri and B. Vitale, On the inversion of Noether's theorem in classical dynamical systems, Am. J. Phys., 40 (1972), 424-429.  doi: 10.1119/1.1986566.

[5]

D. S. Djukic, A procedure for finding first integrals of mechanical systems with gauge-variant Lagrangians, Internat. J. Non-Linear Mech., 8 (1973), 479-488.  doi: 10.1016/0020-7462(73)90039-5.

[6]

D. S. Djukic and B. D. Vujanovic, Noether's theory in classical nonconservative mechanics, Acta Mechanica, 23 (1975), 17-27.  doi: 10.1007/BF01177666.

[7]

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.

[8]

G. Gorni and G. Zampieri, Nonlocal variational constants of motion in dissipative dynamics, Differ. Integral Equ., 30 (2017), 631-640. 

[9]

G. Gorni and G. Zampieri, Nonstandard separation of variables for the Maxwell-Bloch conservative system, São Paulo J. Math. Sci. , published online 27 October 2017. doi: 10.1007/s40863-017-0079-3.

[10]

J. A. Kobussen, On a systematic search for integrals of motion, Helv. Phys. Acta, 53 (1980), 183-200. 

[11]

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

[12]

R. Leone and T. Gourieux, Classical Noether theory with application to the linearly damped particle, European J. Phys. , 36 (2015), 065022, 20pp. doi: 10.1088/0143-0807/36/6/065022.

[13]

F. X. Mei, Lie symmetries and conserved quantities of constrained mechanical systems, Acta Mechanica, 141 (2000), 135-148.  doi: 10.1007/BF01268673.

[14]

J. Moser, Three integrable Hamiltonian systems connected with isospectral deformations, Adv. Math., 16 (1975), 197-220.  doi: 10.1016/0001-8708(75)90151-6.

[15]

W. Sarlet and F. Cantrijn, Generalizations of Noether's theorem in classical mechanics, SIAM Review, 23 (1981), 467-494.  doi: 10.1137/1023098.

[16]

E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4$^{th}$ Edition, Cambridge University Press, New York, 1959.

show all references

References:
[1]

F. T. Arecchi and R. Meucci, Chaos in lasers, Scholarpedia, 3 (2008), 7066. doi: 10.4249/scholarpedia.7066.

[2]

L. Y. Bahar and H. G. Kwatny, Extension of Noether's theorem to constrained non-conservative dynamical systems, Int. J. Non-Linear Mechanics, 22 (1987), 125-138.  doi: 10.1016/0020-7462(87)90015-1.

[3]

F. Calogero, Solutions of the one dimensional $n$-body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys, 12 (1971), 419-436.  doi: 10.1063/1.1665604.

[4]

E. CandottiC. Palmieri and B. Vitale, On the inversion of Noether's theorem in classical dynamical systems, Am. J. Phys., 40 (1972), 424-429.  doi: 10.1119/1.1986566.

[5]

D. S. Djukic, A procedure for finding first integrals of mechanical systems with gauge-variant Lagrangians, Internat. J. Non-Linear Mech., 8 (1973), 479-488.  doi: 10.1016/0020-7462(73)90039-5.

[6]

D. S. Djukic and B. D. Vujanovic, Noether's theory in classical nonconservative mechanics, Acta Mechanica, 23 (1975), 17-27.  doi: 10.1007/BF01177666.

[7]

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.

[8]

G. Gorni and G. Zampieri, Nonlocal variational constants of motion in dissipative dynamics, Differ. Integral Equ., 30 (2017), 631-640. 

[9]

G. Gorni and G. Zampieri, Nonstandard separation of variables for the Maxwell-Bloch conservative system, São Paulo J. Math. Sci. , published online 27 October 2017. doi: 10.1007/s40863-017-0079-3.

[10]

J. A. Kobussen, On a systematic search for integrals of motion, Helv. Phys. Acta, 53 (1980), 183-200. 

[11]

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

[12]

R. Leone and T. Gourieux, Classical Noether theory with application to the linearly damped particle, European J. Phys. , 36 (2015), 065022, 20pp. doi: 10.1088/0143-0807/36/6/065022.

[13]

F. X. Mei, Lie symmetries and conserved quantities of constrained mechanical systems, Acta Mechanica, 141 (2000), 135-148.  doi: 10.1007/BF01268673.

[14]

J. Moser, Three integrable Hamiltonian systems connected with isospectral deformations, Adv. Math., 16 (1975), 197-220.  doi: 10.1016/0001-8708(75)90151-6.

[15]

W. Sarlet and F. Cantrijn, Generalizations of Noether's theorem in classical mechanics, SIAM Review, 23 (1981), 467-494.  doi: 10.1137/1023098.

[16]

E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4$^{th}$ Edition, Cambridge University Press, New York, 1959.

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