SIGMA 1 (2005), 024, 12 pages      nlin.SI/0512012      https://doi.org/10.3842/SIGMA.2005.024

Symmetry Properties of Autonomous Integrating Factors

Sibusiso Moyo ^a and P.G.L. Leach ^b
a) Department of Mathematics, Durban Institute of Technology, PO Box 953, Steve Biko Campus, Durban 4000, Republic of South Africa
^b) School of Mathematical Sciences, Howard College, University of KwaZulu-Natal, Durban 4041, Republic of South Africa

Received September 27, 2005, in final form November 21, 2005; Published online December 05, 2005

Abstract
We study the symmetry properties of autonomous integrating factors from an algebraic point of view. The symmetries are delineated for the resulting integrals treated as equations and symmetries of the integrals treated as functions or configurational invariants. The succession of terms (pattern) is noted. The general pattern for the solution symmetries for equations in the simplest form of maximal order is given and the properties of the associated integrals resulting from this analysis are given.

Key words: autonomous integrating factors; maximal symmetry.

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references

1. Bluman G.W., Kumei S., Symmetries and differential equations, New York, Springer-Verlag, 1989.
2. Cheb-Terrab E.S., Roche A.D., Integrating factors for second order ordinary differential equations, J. Symbolic Comput., 1999, V.27, 501-519, math-ph/0002025.
3. Leach P.G.L., Bouquet S.É., Symmetries and integrating factors, J. Nonlinear Math. Phys., 2002, V.9, suppl. 2, 73-91.
4. Abraham-Shrauner B., Hidden symmetries, first integrals and reduction of order of nonlinear ordinary differential equations, J. Nonlinear Math. Phys., 2002, V.9, suppl. 2, 1-9.
5. Head A., LIE, a PC program for Lie analysis of differential equations, Comput. Phys. Comm., 1993, V.77, 241-248.
6. Ermakov V., Second-order differential equations. Conditions of complete integrability, Universitetskie Izvestiya, Kiev, 1880, N 9, 1-25 (translated by A.O. Harin).
7. Pinney E., The nonlinear differential equation y"(x) + p(x) y + cy^-3 = 0, Proc. Amer. Math. Soc., 1950, V.1, 681.
8. Lewis H.R., Classical and quantum systems with time-dependent harmonic oscillator-type Hamiltonians, Phys. Rev. Lett., 1967, V.18, 510-512.
9. Lewis H.R.Jr., Motion of a time-dependent harmonic oscillator and of a charged particle in a time-dependent, axially symmetric, electromagnetic field, Phys. Rev., 1968, V.172, 1313-1315.
10. Mahomed F.M., Leach P.G.L., Symmetry Lie algebras of nth order ordinary differential equations, J. Math. Anal. Appl., 1990, V.151, 80-107.
11. Mubarakzyanov G.M., On solvable Lie algebras, Izv. Vys. Uchebn. Zaved. Matematika, 1963, N 1 (32), 114-123.
12. Mubarakzyanov G.M., Classification of real structures of five-dimensional Lie algebras, Izv. Vys. Uchebn. Zaved. Matematika, 1963, N 3 (34), 99-106.
13. Mubarakzyanov G.M., Classification of solvable Lie algebras of sixth order with a non-nilpotent basis element, Izv. Vys. Uchebn. Zaved. Matematika, 1963, N 4 (35), 104-116.
14. Patera, J., Sharp R.T., Winternitz P., Invariants of real low dimension Lie algebras, J. Math. Phys., 1976, V.17, 986-994.
15. Flessas G.P., Govinder K.S., Leach P.G.L., Remarks on the symmetry Lie algebras of first integrals of scalar third order ordinary differential equations with maximal symmetry, Bull. Greek Math. Soc., 1994, V.36, 63-79.
16. Flessas G.P., Govinder K.S., Leach P.G.L., Characterisation of the algebraic properties of first integrals of scalar ordinary differential equations of maximal symmetry, J. Math. Anal. Appl. 1997, V.212, 349-374.
17. Moyo S., Leach P.G.L., Exceptional properties of second and third order ordinary differential equations of maximal symmetry, J. Math. Anal. Appl., 2000, V.252, 840-863.