Symmetries of First Integrals and Their Associated Differential Equations

Journal of Mathematical Analysis and Applications

Volumn 235, Issue 1, 1999, Pages 58-83

  Leach, P G L ^a   Govinder, K S ^b   Abraham Shrauner, B ^c  

^a UNIVERSITY OF THE AEGEAN   (Greece)

^b UNIVERSITY OF KWAZULU NATAL   (South Africa)

^c WASHINGTON UNIVERSITY IN ST LOUIS   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0002548675     PISSN: 0022247X     EISSN: None     Source Type: Journal    
DOI: 10.1006/jmaa.1999.6361     Document Type: Article

References (34)

1. Abraham-Shrauner B. Hidden symmetries and linearization of the modified Painlevé-Ince equation. J. Math. Phy. 34:1993;4809-4816.
2. Abraham-Shrauner Barbara. Hidden symmetries and nonlocal group generators for ordinary differential equations. Ames W. F. Proceedings of the 14th IMACS World Congress on Computational and Applied Mathematics. 1994;1-14 Georgia Institute of Technology, Atlanta.
3. Abraham-Shrauner B., Govinder K. S., Leach P. G. L. Integration of second order ordinary differential equations not possessing Lie point symmetries. Phys. Lett. A. 209:1995;169-174.
4. Abraham-Shrauner B., Guo A. Hidden symmetries associated with the projective group of nonlinear first-order ordinary differential equations. J. Physics A. 25:1993;5597-5608.
5. B. Abraham-Shrauner, G. Ratcliff, K. S. Govinder, and, P. G. L. Leach, Linearisation of third order ordinary differential equations by contact transformations, preprint, Department of Electrical Engineering, Washington University, 1995.
6. Dickson L. E. Differential equations from the group standpoint. Ann. of Math. 25:1925;287-378.
7. G. P. Flessas, K. S. Govinder, and, P. G. L. Leach, Remarks on the symmetry Lie algebras of first integrals of scalar third order ordinary differential equations with maximal symmetry, Bull. Greek Math. Soc, to appear.
8. González-Gascón F., González-López A. Newtonian systems of differential equations integrable via quadratures, with trivial group of point symmetries. Phys. Lett. A. 129:1988;153-156.
9. Govinder K. S., Leach P. G. L. The algebraic structures of the first integrals of third order linear equations. J. Math. Anal. Appl. 193:1995;114-133.
10. Govinder K. S., Leach P. G. L. Noetherian integrals via nonlocal transformation. Phy. Lett. A. 201:1995;91-94.
11. Govinder K. S., Leach P. G. L. On the determination of nonlocal symmetries. J. Phys. A: Mathematical and General. 28:1995;5349-5359.
12. Govinder K. S., Leach P. G. L. An elementary demonstration of the existence of sℓ(3, R) symmetry for all second order linear ordinary differential equations. SIAM Rev. 40:1998;945-946.
13. Hall L. S. A theory of exact and approximate configurational invariants. Phys. D. 8:1983;90-105.
14. Head A. K. LIE, a PC program for Lie analysis of differential equations. Comput. Phys. Comm. 77:1993;241-248.
15. Hsu L., Kamran N. Symmetries of second-order ordinary differential equations and Elie Cartan's method of equivalence. Lett. Math. Phys. 15:1988;91-99.
16. Ince E. L. Ordinary Differential Equations. 1927;Longmans, Green & Co. London.
17. Kamke E. Teubner B. G. Differentialgleichungen: Lösungsmethoden und Lösungen, Band I, Gewöhnliche Differentialgleichungen. 1983.
18. Kummer E. E. De generali quadam æquatione differentiali tertii ordinis. Journal für die Reine und Angewandte Mathematik. 100:1887;1-9.
19. Leach P. G. L., Feix M. R., Bouquet S. Analysis and solution of a nonlinear second order differential equation through rescaling and through a dynamical point of view. J. Math. Phys. 29:1988;2563-2569.
20. Leach P. G. L., Mahomed F. M. Maximal subalgebra associated with a first integral of a system possessing sℓ(3, R) algebra. J. Math. Phys. 29:1988;1807-1813.
21. 3 = 0. J. Phys. A. 26:1994;5017-5024.
22. Lie Sophus. Differentialgleichungen. 1967;Chelsea, New York.
23. Lie Sophus. Geometrie der Berührungstransformationen. 1977;Chelsea, New York.
24. Liouville J. Sur le développement des fonctions ou parties de fonctions en séries dont les divers termes sont assujettis à satisfaire à une même équatione différentielle du second ordre contenant un paramètre variable. J. Math. Pures Appliq. (9). II:1837;16-35.
25. F. M. Mahomed, Classification of Nonlinear Ordinary Differential Equations According to Their Symmetries, MSc. dissertation, University of the Witwatersrand, Johannesburg, 1986.
26. Mahomed F. M., Leach P. G. L. The linear symmetries of a nonlinear differential equation. Quæstiones Math. 8:1985;241-274.
27. Mahomed F. M., Leach P. G. L. Lie algebras associated with scalar second-order ordinary differential equations. J. Math. Phys. 30:1989;2770-2777.
28. Mahomed F. M., Leach P. G. L. Symmetry Lie algebras of nth order ordinary differential equations. J. Math. Anal. Appl. 151:1990;80-107.
29. Olver Peter J. Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics. 107:1993;Springer-Verlag, New York.
30. Patera J., Sharp R. T., Winternitz P., Zassenhaus H. Invariants of real low dimension Lie algebras. J. Math. Phys. 17:1976;986-994.
31. Patera J., Winternitz P. Continuous subgroups of the fundamental groups of physics. I. General method and the Poincaré group. J. Math. Phys. 16:1975;1597-1614.
32. Patera J., Winternitz P. Subalgebras of real three- and four-dimensional Lie algebras. J. Math. Phys. 18:1977;1449-1455.
33. Sarlet W., Leach P. G. L., Cantrijn F. Exact versus configurational invariants and a weak form of complete integrability. Phys. D. 17:1985;87-98.
34. F. E. Vawda, An Application of Lie Analysis to Classical Mechanics, MSc. dissertation, University of the Witwatersrand, Johannesburg, 1994.