Abstract
The problem of differential equation systems admitting a nonlinear superposition principle is analyzed from a geometric perspective. We show how it is possible to reduce the problem of finding the general solution of such a differential equation system defined by a Lie group G to a pair of simpler problems, one in a subgroup H and the other on a homogeneous space. The theory is illustrated with several examples and applications.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anderson, R. L. and Davison, S. M.: A generalization of Lie's 'counting' theorem for second-order ordinary differential equations, J. Math. Anal. Appl. 48 (1974), 301–315.
Cariñena, J. F., Marmo, G. and Nasarre, J.: The nonlinear superposition principle and the Wei-Norman method, Internat. J. Modern Phys. A 13 (1998), 3601–3627.
Cariñena, J. F., Marmo, G., Perelomov, A. M. and Rañada, M. F.: Related operators and exact solutions of Schrödinger equations, Internat. J. Modern Phys. A 13 (1998), 4913–4929.
Cariñena, J. F. and Ramos, A.: Integrability of Riccati equation from a group theoretical viewpoint, Internat. J. Modern Phys. A 14 (1999), 1935–1951.
Wei, J. and Norman, E.: Lie algebraic solution of linear differential equations, J. Math. Phys. 4 (1963), 575–581.
Wei, J. and Norman, E.: On global representations of the solutions of linear differential equations as a product of exponentials, Mem. Amer. Math. Soc. 15 (1964), 327–334.
Lie, S.: Vorlesungen über continuierliche Gruppen mit geometrischen und anderen Anwendungen (revised and edited by Dr G. Scheffers), B.G. Teubner, Leipzig, 1893.
Libermann, P. and Marle, Ch.-M.: Symplectic Geometry and Analytical Mechanics, D. Reidel, Dordrecht, 1987.
Winternitz, P.: Lie groups and solutions of nonlinear differential equations, In: K. B. Wolf (ed.), Nonlinear Phenomena, Lecture Notes in Phys. 189, Springer, New York, 1983.
Winternitz, P.: Comments on superposition rules for nonlinear coupled first-order differential equations, J. Math. Phys. 25 (1984), 2149–2150.
Shnider, S. and Winternitz, P.: Classification of systems of ordinary differential equations with superposition principles, J. Math. Phys. 25 (1984), 3155–3165.
Jacobson, N.: Lie Algebras, Interscience Publishers, New York, 1961.
Humphreys, J. E.: Introduction to Lie Algebras and Representation Theory, Springer, New York, 1972.
Vessiot, E.: Sur une classe d'équations différentielles, Ann. Sci. École Norm. Sup. 10 (1893), 53.
Guldberg, A.: Sur les équations différentielles que possedent un système fundamental d'intégrales, C.R. Acad. Sci. Paris 116 (1893), 964.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cariñena, J.F., Grabowski, J. & Ramos, A. Reduction of Time-Dependent Systems Admitting a Superposition Principle. Acta Applicandae Mathematicae 66, 67–87 (2001). https://doi.org/10.1023/A:1010743114995
Issue Date:
DOI: https://doi.org/10.1023/A:1010743114995