87
Views
2
CrossRef citations to date
0
Altmetric
Original Articles

Jacobi’s Three-Body System Moves like a Free Particle

Pages 499-506 | Published online: 21 Jan 2013

References

  • Ames W F Nonlinear Partial Differential Equations in Engineering, Vol. 2, Academic Press, New York, 1972
  • Anderson R M The role of mathematical models in the study of HIV transmission and the epidemiology of AIDS, J. of AIDS 1 (1988), 240–255
  • Bianchi L Lezioni sulla teoria dei gruppi continui finiti di trasformazioni, Enrico Spoerri, Pisa, 1918
  • Bluman G W Cole J D Similarity Methods for Differential Equations, Springer-Verlag, Berlin, 1974
  • Bluman G W Kumei S Symmetries and Differential Equations, Springer-Verlag, Berlin, 1989
  • Calogero F Solution of a three-body problem in one dimension, J. Math. Phys. 10 (1969), 2191–2196
  • Calogero F Motion of poles and zeros of special solutions of nonlinear and linear partial differential equations and related “solvable” many-body problems, Nuovo Cimento B 43 (1978), 177–241
  • Calogero F Classical Many-Body Problems Amenable to Exact Treatments, Springer-Verlag, Berlin, 2001
  • Calogero F The neatest many-body problem amenable to exact treatments (a “goldfish”?), Physica D 152–153 (2001), 78–84
  • Clarkson P A Olver P J Symmetry and the Chazy equation, J. Differential Equations 124 (1996), 225–246
  • Edwards M Nucci M C Application of Lie group analysis to a core group model for sexually transmitted diseases, Dipartimento di Matematica e Informatica, Università di Perugia, Preprint RT2003–6, 2003
  • Hadeler K P Castillo-Chavez C A core group model for disease transmission, Math. Biosci. 128 (1995), 41–55
  • Hawkins T The Emergence of the Theory of Lie Groups: An Essay in the History of Mathematics 1869–1926, Springer-Verlag, Berlin, 2000
  • Hawkins T Jacobi and the birth of Lie’s theory of groups, Arch. Hist. Exact Sciences 42 (1991), 187–278
  • Hill J M Differential Equations and Group Methods for Scientists and Engineers, CRC Press, Boca Raton, 1992
  • Hydon P E Symmetry Methods for Differential Equations: A Beginner’s Guide, Cambridge University Press, Cambridge, 2000
  • Ibragimov N H Opyt Gruppovogo Analiza Obyknovennykh Differentsial’nykh Uravnenii, Znanie, Moscow, 1991
  • Ibragimov N H Nucci M C Integration of third order ordinary differential equations by Lie’s method: equations admitting three-dimensional Lie algebras, Lie Groups and Their Applications 1 (1994), 49–64
  • Ibragimov N H (ed.), CRC Handbook of Lie Group Analysis of Differential Equations, Vol. I: Symmetries, Exact Solutions, and Conservation Laws, CRC Press, Boca Raton, 1994
  • Ibragimov N H (ed.), CRC Handbook of Lie Group Analysis of Differential Equations, Vol. II: Applications in Engineering and Physical Sciences, CRC Press, Boca Raton, 1995
  • Ibragimov N H Elementary Lie Group Analysis and Ordinary Differential Equations, Wiley, New York, 1999
  • Jacobi C G J Theoria novi multiplicatoris systemati æquationum differentialium vulgarium applicandi, J. Reine Angew. Math. 29 (1845), 333–376
  • Jacobi C G J Problema trium corporum mutuis attractionibus cubis distantiarum inverse proportionalibus recta linea se moventium, in Gesammelte Werke, t. 4, Berlin, 1866, 533–539
  • Koenigsberger L Carl Gustav Jacob Jacobi, Teubner, Leipzig, 1904
  • Kowalevski S Sur le problème de la rotation d’un corps solide autour d’un point fixe, Acta Math. 12 (1889), 177–232
  • Leach P G L Nucci M C Reduction of the classical MICZ-Kepler problem to a two-dimensional linear isotropic harmonic oscillator, J. Math. Phys. 45 (2004), 3590–3604
  • Lie S Theorie der Transformationsgruppen. Erster Abschnitt, Teubner, Leipzig, 1888
  • Lie S Theorie der Transformationsgruppen. Zweiter Abschnitt, Teubner, Leipzig, 1890
  • Lie S Theorie der Transformationsgruppen. Dritter und Letzter Abschnitt, Teubner, Leipzig, 1893
  • Lie S Geometrie der Berührungstransformationen, Teubner, Leipzig, 1896
  • Lie S Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen, Teubner, Leipzig, 1912
  • Lorenz E N Deterministic nonperiodic flow, J. Atmos. Sci. 20 (1963), 130–141
  • Marcelli M Nucci M C Lie point symmetries and first integrals: the Kowalevsky top, J. Math. Phys. 44 (2003), 2111–2132
  • Noether E Invariante Variationsprobleme, Nachr d König Gesellsch d Wiss zu Göttingen, Math-phys Klasse (1918), 235–257
  • Nucci M C Interactive REDUCE programs for calculating classical, non-classical and Lie-Bäcklund symmetries of differential equations, Preprint Georgia Tech Math 062090–051, 1990
  • Nucci M C Interactive REDUCE programs for calculating Lie point, non-classical, Lie-Bäcklund, and approximate symmetries of differential equations: manual and floppy disk, in CRC Handbook of Lie Group Analysis of Differential Equations. Vol. 3: New Trends, Editor: Ibragimov N H, CRC Press, Boca Raton, 1996, 415–481
  • Nucci M C The complete Kepler group can be derived by Lie group analysis, J. Math. Phys. 37 (1996), 1772–1775
  • Nucci M C Lorenz integrable system moves à la Poinsot, J. Math. Phys. 44 (2003), 4107– 4118
  • Nucci M C Calogero’s “goldfish” is indeed a school of free particles, J. Phys. A: Math. Gen. 37 (2004), 11391–11400
  • Olver P J Applications of Lie Groups to Differential Equations, Springer-Verlag, Berlin, 1986
  • Ovsjannikov L V Group Analysis of Differential Equations, Academic Press, New York, 1982
  • Rogers C Ames W F Nonlinear Boundary Value Problems in Science and Engineering, Academic Press, New York, 1989
  • Stephani H Differential Equations. Their Solution Using Symmetries, Cambridge University Press, Cambridge, 1989
  • Torrisi V Nucci M C Application of Lie group analysis to a mathematical model which describes HIV transmission, in The Geometrical Study of Differential Equations, Editors: Leslie J A and Hobart T P, AMS, Providence, 2001, 31–40

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.