This paper describes a new algorithm for determining all discrete contact symmetries of any differential equation whose Lie contact symmetries are known. The method is constructive and is easy to use. It is based upon the observation that the adjoint action of any contact symmetry is an automorphism of the Lie algebra of generators of Lie contact symmetries. Consequently, all contact symmetries...

For a linear ordinary differential equation the Lie algebra of its infinitesimal Lie symmetries is compared with its differential Galois group. For this purpose an algebraic formulation of Lie symmetries is developed. It turns out that there is no direct relation between the two above objects. In connection with this a new algorithm for computing the Lie symmetries of a linear ordinary differen...

A review of the role of symmetries in solving differential equations is presented. After showing some recent results on the application of classical Lie point symmetries to problems in fluid draining, meteorology, and epidemiology of AIDS, the nonclassical symmetries method is presented. Finally, it is shown that iterations of the nonclassical symmetries method yield new nonlinear equations, wh...

It is shown that the Lie algebra of Noether symmetries for the Lagrangian minimizing an n − 1 area enclosing a constant n-volume in a Euclidean space is so n ⊕sR and in a space of constant curvature the Lie algebra is so n . Furthermore, if the space has one section of constant curvature of dimension n1, another of n2, and so on to nk and one of zero curvature of dimension m, with n ≥ ∑kj 1 nj ...

The discrete symmetries P, C and T are discussed in terms of Lie algebra extensions of the Poincare Lie algebra. This formulation leads to certain problems of Lie algebra theory which will be presented in this and succeeding papers. (To be submitted to Communications in Mathematical Physics) * Work supported by the U. S. Atomic Energy Commission.

The classical reduction of order for scalar ordinary differential equations (ODEs) fails for a system of ODEs. We prove a constructive result for the reduction of order for a system of ODEs that admits a solvable Lie algebra of point symmetries. Applications are given for the case of a system of two second-order ODEs which admits a solvable four-dimensional Lie algebra of point symmetries.

This article provides us with a unifying classification of the conformal infinitesimal symmetries of non-relativistic Newton-Cartan spacetime. The Lie algebras of non-relativistic conformal transformations are introduced via the Galilei structure. They form a family of infinite-dimensional Lie algebras labeled by a rational “dynamical exponent”, z. The Schrödinger-Virasoro algebra of Henkel et ...

This article provides us with a unifying classification of the conformal infinitesimal symmetries of non-relativistic Newton-Cartan spacetime. The Lie algebras of non-relativistic conformal transformations are introduced via the Galilei structure. They form a family of infinite-dimensional Lie algebras labeled by a rational “dynamical exponent”, z. The Schrödinger-Virasoro algebra of Henkel et ...

In this article, we generalize the concept of the Lie algebra of vector fields to the set of smooth sections of a T-bundle which is by definition a canonical generalization of the concept of a tangent bundle. We define a Lie bracket multiplication on this set so that it becomes a Lie algebra. In the particular case of tangent bundles this Lie algebra coincides with the Lie algebra of vector fie...

By using the self-dual Yang-Mills (SDYM) equation as an example, we study a method for relating symmetries and recursion operators of two partial differential equations connected to each other by non-auto-Backlund transformation. We prove Lie-algebra isomorphism between SDYM those potential (PSDYM) equation, describe construction these systems. Using certain known aspects PSDYM symmetry algebra...