Properties of some autonomous equations

invariant under homogeneity symmetries

M.R.Feix, C.Géronimi, P.G.L.Leach

MAPMO/URA CNRS 1803 - Université d'Orléans

Département mathématiques BP 6759

45067 La Source, Orléans Cedex 2, France

geronimi.claude@wanadoo.frљљљљљљљљ leachp@nu.ac.za

 

Although not common, third- and fourth-order ordinary differential equations can be found in Physics (mostly in cosmological problems). For systems exhibiting time translation and the two homogeneity symmetries the third-order ordinary differential equation can be completely integrated while that of the fourth order can be reduced to a first-order equation in the two variables љand . Moreover a second-order ordinary differential equation possessing time translation and rescaling symmetries can be transformed into a third-order ordinary differential equation possessing these three symmetries through a Riccati transformation and this last equation may be more tractable. These ideas are illustrated through the treatment of different examples.

 

 

Marc Roy Feix, Dr., longtime member of the CNRS and Director of its laboratory, Physique Mathématique Modélisation et Simulation, in Orléans and subsequently within the Département mathématiques, Université d'Orléans; active in plasma physics, differential equations and all sorts of modelling and simulation problems.

Claude Géronimi, Graduate of the Université d'Orléans; student of the noted plasma physicist, Marc Feix; interested in problems relating to the symmetries and singularities of differential equations.

Peter Gavin Lawrence Leach, Dr., Graduate of the University of Melbourne and La Trobe University in Australia and the University of Natal in South Africa; Professor of Mathematics of the University of Natal, Durban; formerly Professor of Applied Mathematics, University of the Witwatersrand, Johannesburg; active in the applications of symmetry and singularity analyses to differential equations arising in Mechanics, Quantum Mechanics, Cosmology, Mathematical Economics and Mathematical Biosciences; also interested in problems of molecular structure.