Geodesic
Jacobi's last multiplier, Lie symmetries, and hidden linearity: “Goldfishes" galore
Teoretičeskaâ i matematičeskaâ fizika, Tome 151 (2007) no. 3, pp. 495-509 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In addition to the reduction method, we present a novel application of Jacobi's last multiplier for finding Lie symmetries of ordinary differential equations algorithmically. These methods and Lie symmetries allow unveiling the hidden linearity of certain nonlinear equations that are relevant in physics. We consider the Einstein–Yang–Mills equations and Calogero's many-body problem in the plane as examples.
Keywords: Lie group analysis, first integral
Mots-clés : Jacobi's last multiplier. 
@article{TMF_2007_151_3_a14,
     author = {M. C. Nucci},
     title = {Jacobi's last multiplier, {Lie} symmetries, and hidden linearity: {{\textquotedblleft}Goldfishes"} galore},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {495--509},
     year = {2007},
     volume = {151},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2007_151_3_a14/}
}
TY  - JOUR
AU  - M. C. Nucci
TI  - Jacobi's last multiplier, Lie symmetries, and hidden linearity: “Goldfishes" galore
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2007
SP  - 495
EP  - 509
VL  - 151
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_2007_151_3_a14/
LA  - ru
ID  - TMF_2007_151_3_a14
ER  - 
%0 Journal Article
%A M. C. Nucci
%T Jacobi's last multiplier, Lie symmetries, and hidden linearity: “Goldfishes" galore
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2007
%P 495-509
%V 151
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2007_151_3_a14/
%G ru
%F TMF_2007_151_3_a14
M. C. Nucci. Jacobi's last multiplier, Lie symmetries, and hidden linearity: “Goldfishes" galore. Teoretičeskaâ i matematičeskaâ fizika, Tome 151 (2007) no. 3, pp. 495-509. http://geodesic.mathdoc.fr/item/TMF_2007_151_3_a14/

[1] M. C. Nucci, J. Math. Phys., 37 (1996), 1772–1775 | DOI | MR | Zbl

[2] V. Torrisi, M. C. Nucci, “Application of Lie group analysis to a mathematical model which describes HIV transmission”, The Geometrical Study of Differential Equations (Washington, DC, USA, 2000), Contemp. Math., 285, eds. J. A. Leslie, T. P. Hobart, Amer. Math. Soc., Providence, RI, 2001, 11–20 | DOI | MR | Zbl

[3] M. C. Nucci, P. G. L. Leach, J. Math. Phys., 42 (2001), 746–764 | DOI | MR | Zbl

[4] M. C. Nucci, J. Math. Phys., 44 (2003), 4107–4118 | DOI | MR | Zbl

[5] P. G. L. Leach, M. C. Nucci, J. Math. Phys., 45 (2004), 3590–3604 | DOI | MR | Zbl

[6] M. C. Nucci, J. Phys. A, 37 (2004), 11391–11400 | DOI | MR | Zbl

[7] M. Edwards, M. C. Nucci, J. Nonlin. Math. Phys., 13 (2006), 211–230 | DOI | MR

[8] A. Gradassi, M. C. Nucci, J. Math. Anal. Appl., 333 (2007), 274–294 | DOI | MR | Zbl

[9] A. Gradassi, M. C. Nucci, Integrability in a variant of the three-species Lotka–Volterra model (to appear) | Zbl

[10] M. Marcelli, M. C. Nucci, J. Math. Phys., 44 (2003), 2111–2132 | DOI | MR | Zbl

[11] E. Noether, “Invariante Variationsprobleme”, Nachr. v. d. Ges. d. Wiss. zu Göttingen, 2 (1918), 235–257 | Zbl

[12] M. C. Nucci, J. Nonlin. Math. Phys., 12 (2005), 284–304 | DOI | MR | Zbl

[13] C. G. J. Jacobi, Giornale Arcadico di Scienze, Lettere ed Arti, 99 (1844), 129–146 | MR

[14] C. G. J. Jacobi, J. Reine Angew. Math., 27 (1844), 199–268 | DOI | MR | Zbl

[15] C. G. J. Jacobi, J. Reine Angew. Math., 29 (1845), 213–279 ; 333–376 | DOI | MR | Zbl | Zbl

[16] K. Yakobi, Lektsii po dinamike, Gostekhizdat, L., 1936

[17] S. Lie, “Verallgemeinerung und neue Verwwandlung der Jacobischen Multiplicator-Théorie”, Forh. Christiania, 1874, 255–274 | Zbl

[18] S. Lie, Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen, Teubner, Leipzig, 1912 | Zbl

[19] L. Bianchi, Lezioni sulla teoria dei gruppi continui finiti di trasformazioni, Spoerri, Pisa, 1918 | Zbl

[20] E. Uitteker, Analiticheskaya dinamika, Izd-vo UdGU, Izhevsk, 1999 | MR | Zbl | Zbl

[21] E. Kamke, Spravochnik po obyknovennym differentsialnym uravneniyam, Nauka, M., 1976 | MR | MR | Zbl | Zbl | Zbl

[22] A. González-López, J. Math. Phys., 29 (1988), 1097–1105 | DOI | MR | Zbl

[23] J. Llibre, C. Valls, J. Phys. A, 38 (2005), 8155–8168 | DOI | MR | Zbl

[24] F. Calogero, Classical Many-Body Problems Amenable to Exact Treatments, Lect. Notes Phys. New Ser. m: Monogr., 66, Springer, Berlin, 2001 | DOI | MR | Zbl

[25] M. C. Nucci, “Interactive REDUCE programs for calculating Lie point, non-classical, Lie–Bäcklund, and approximate symmetries of differential equations: manual and floppy disk”, CRC Handbook of Lie Group Analysis of Differential Equations, v. 3. New Trends in Theoretical Developments and Computational Methods, ed. N. H. Ibragimov, CRC Press, Boca Raton, 1996, 415–481 | MR | Zbl

[26] F. Calogero, Nuovo Cimento B, 43 (1978), 177–241 | DOI | MR

[27] F. Calogero, Physica D, 152–153 (2001), 78–84 | DOI | MR | Zbl

[28] V. E. Zakharov, “On the dressing method”, Inverse Methods in Action (Montpellier, 1989), Inverse Probl. Theor. Imaging, ed. P. C. Sabatier, Springer, Berlin, 1990, 602–623 | DOI | MR | Zbl

[29] M. Bruschi, F. Calogero, J. Math. Phys., 47:10 (2006), 102701 | DOI | MR | Zbl

[30] F. Kalodzhero, TMF, 133 (2002), 149–159 ; Erratum, 134:1 (2003), 160 | DOI | MR | DOI | MR

[31] A. Guillot, Comm. Math. Phys., 256 (2005), 181–194 | DOI | MR | Zbl

[32] M. C. Nucci, “An ODE system connected to isospectral beams (Solution to problem 2005-4)”, Electron. J. Diff. Eqns. Problem Section: 2005-4, 2006; http://math.uc.edu/ode/odesols/p20054.pdf

[33] M. C. Nucci, P. G. L. Leach, J. Phys. A, 37 (2004), 7743–7753 | DOI | MR | Zbl

[34] M. K. Nuchchi, TMF, 144 (2005), 394–404 | DOI | MR

[35] M. C. Nucci, P. G. L. Leach, J. Math. Phys., 48 (2007), 013514 | DOI | MR | Zbl