Jacobi’s Last Multiplier and the Complete Symmetry Group of the Euler—Poinsot System
Tóm tắt
The symmetry approach to the determination of Jacobi’s last multiplier is inverted to provide a source of additional symmetries for the Euler—Poinsot system. These additional symmetries are nonlocal. They provide the symmetries for the representation of the complete symmetry group of the system.
Tài liệu tham khảo
Andriopoulos K, Leach P G L and Flessas G P, Complete Symmetry Groups of Ordinary Differential Equations and Their Integrals: Some Basic Considerations, J. Math. Anal. Appl. 262 (2001), 256–273.
Andriopoulos K and Leach P G L, The Economy of Complete Symmetry Groups for Linear Higher Dimensional Systems, J. Nonlin. Math. Phys. 9, Suppl. 2 (2002), 10–23.
Bianchi L, Lezione sulla teoria dei gruppi continui finiti di transformazioni, Enrico Spoerri, Pisa, 1918.
Cohen A, An Introduction to the Lie Theory of One-Parameter Groups, Stechert, New York, reprinted 1931.
Dickson LE, Differential Equations from the Group Standpoint, Ann. Math. 25 (1925), 287–378.
Eisenhart L P, Continuous Groups of Transformations, Dover, New York, 1961.
Ermakov V, Second Order Differential Equations. Conditions of Complete Integrability, Univ. Izvestia Kiev, Ser III 9 (1880), 1–25 (translated by A O Harin).
Euler L, Theoria motus corporum solidorum seu rigidorum ex primis nostræcognitionis principiis stabilita et ad omnes motus, qui in huiusmodi corpora cadere possunt, accommodata, Rostochii et Gryphiswaldiae, Litteris et Impensis Röse AF, 1765.
Hawkins T, Jacobi and the Birth of Lie’s Theory of Groups, Arch. Hist. Exact Sciences 42 (1991), 187–278.
Jacobi C G J, Sur un théorème de Poisson, Comptes Rendus Acad. Sci. Paris 11 (1840), 529–530.
Jacobi C G J, Adressé à M. le president de l’Academie des Sciences, J. de Math. V (1840), 350–355 (including editor’s notes).
Jacobi C G J, Sul principio dell’ultimo moltiplicatore, e suo uso come nuovo principio generale di meccanica, Giornale arcadico di scienze, lettere ed arti Tomo 99 (1844), 129–146.
Jacobi C G J, Theoria novi multiplicatoris systemati æquationum differentialum vulgarium applicandi: Pars I, J. für Math. 27 (1844), 199.
Jacobi C G J, Theoria novi multiplicatoris systemati æquationum differentialum vulgarium applicandi: Pars II, J. für Math. 29 (1845), 213.
Koenigsberger L, Carl Gustav Jacob Jacobi, B T Teubner, Leipzig, 1904.
Krause J, On the Complete Symmetry Group of the Classical Kepler System, J. Math. Phys. 35 (1994), 5734–5748.
Leach P G L, Cotsakis S and Flessas G P, Symmetry, Singularity and Integrability in Complex Dynamics: I The Reduction Problem, J. Nonlin. Math. Phys. 7 (2000), 445–479.
Lie S, Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen, B T Teubner, Leipzig, 1912.
Mahomed F M and Leach P G L, Lie Algebras Associated with Scalar Second Order Ordinary Differential Equations, J. Math. Phys. 30 (1989), 2770–2775.
Marcelli M and Nucci M C, Lie Point Symmetries and First Integrals: the Kowalevsky top, Preprint, RT 2002-1, Dipartimento di Matematica e Informatica, Università di Perugia, 2002.
Nucci M C, Interactive REDUCE Programs for Calculating Classical, Non-Classical and Lie–Bäcklund Symmetries of Differential Equations, Preprint, GT Math: 0620902–051, Department of Mathematics, Georgia Institute of Technology, Atlanta, Georgia, USA, 1990.
Nucci M C, Interactive REDUCE Programs for Calculating Lie Point, Nonclassical, 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 in Theoretical Developments and Computational Methods, 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, Preprint, RT 2002-15, Dipartimento di Matematica e Informatica, Università di Perugia, 2002.
Nucci M C and Leach P G L, The Harmony in the Kepler and Related Problems, J. Math. Phys. 42 (2001), 746–764.
Nucci M C, Andriopoulos K and Leach P G L, The Ermanno-Bernoulli Constants and Representations of the Complete Symmetry Group of the Kepler Problem, Preprint, School of Mathematical and Statistical Sciences, University of Natal, Durban 4041, Republic of South Africa, 2002.
Pinney E, The Nonlinear Differential Equation y″(x)+p(x)y+cy−3 = 0, Proc. Amer. Math. Soc. 1 (1950), 681.
Poinsot L, Théorie nouvelle de la rotations des corps, Paris, 1834 (republished in J. Math. Pures Appl. 16 (1851), 289–336).
Poisson S, Mémoire sur la variation des constantes arbitraires dans les questions de mécanique, J. Ecole Poly 15 (1809), 266–344.