Nonlocal interpretation of λ\lambda-variational symmetry-reduction method

In this paper we give a geometric interpretation of a reduction method based on the so called $\lambda$-variational symmetry (C. Muriel, J.L. Romero and P. Olver 2006 \emph{Variational $C^{\infty}$-symmetries and Euler-Lagrange equations} J. Differential equations \textbf{222} 164-184). In general this allows only a partial reduction but it is particularly suitable for the reduction of variational ODEs with a lack of computable local symmetries. We show that this method is better understood as a nonlocal symmetry-reduction

On the geometry of twisted symmetries: Gauging and coverings

We consider the theory of twisted symmetries of differential equations, in particular \u3bb and \u3bc-symmetries, and discuss their geometrical content. We focus on their interpretation in terms of gauge transformations on the one hand, and of coverings on the other one

Local and nonlocal solvable structures in ODEs reduction

Solvable structures, likewise solvable algebras of local symmetries, can be used to integrate scalar ODEs by quadratures. Solvable structures, however, are particularly suitable for the integration of ODEs with a lack of local symmetries. In fact, under regularity assumptions, any given ODE always admits solvable structures even though finding them in general could be a very difficult task. In practice a noteworthy simplification may come by computing solvable structures which are adapted to some admitted symmetry algebra. In this paper we consider solvable structures adapted to local and nonlocal symmetry algebras of any order (i.e., classical and higher). In particular we introduce the notion of nonlocal solvable structure

Nonlocal aspects of λ\lambda-symmetries and ODEs reduction

A reduction method of ODEs not possessing Lie point symmetries makes use of the so called $\lambda$-symmetries (C. Muriel and J. L. Romero, \emph{IMA J. Appl. Math.} \textbf{66}, 111-125, 2001). The notion of covering for an ODE $\mathcal{Y}$ is used here to recover $\lambda$-symmetries of $\mathcal{Y}$ as nonlocal symmetries. In this framework, by embedding $\mathcal{Y}$ into a suitable system $\mathcal{Y}^{\prime}$ determined by the function $\lambda$, any $\lambda$-symmetry of $\mathcal{Y}$ can be recovered by a local symmetry of $\mathcal{Y}^{\prime}$. As a consequence, the reduction method of Muriel and Romero follows from the standard method of reduction by differential invariants applied to $\mathcal{Y}^{\prime}$.Comment: 13 page

Noether theorem for mu-symmetries

We give a version of Noether theorem adapted to the framework of mu-symmetries; this extends to such case recent work by Muriel, Romero and Olver in the framework of lambda-symmetries, and connects mu-symmetries of a Lagrangian to a suitably modified conservation law. In some cases this "mu-conservation law'' actually reduces to a standard one; we also note a relation between mu-symmetries and conditional invariants. We also consider the case where the variational principle is itself formulated as requiring vanishing variation under mu-prolonged variation fields, leading to modified Euler-Lagrange equations. In this setting mu-symmetries of the Lagrangian correspond to standard conservation laws as in the standard Noether theorem. We finally propose some applications and examples.Comment: 28 pages, to appear in J. Phys.

Nonlocal interpretation of [lambda]-symmetries

A reduction method of ODEs not possessing Lie point symmetries makes use of the so called $\lambda$-symmetries. These symmetries are not standard symmetries, nevertheless for any given $\lambda$-symmetry of an ODE $\mathcal{Y}$ one can always reconstruct nonlocal symmetries of $\mathcal{Y}$% . As a consequence, using these nonlocal symmetries, the $\lambda$-symmetry reduction method follows from the standard method of symmetry reduction by differential invariants

Applications of solvable structures to the nonlocal symmetry-reduction of ODEs

An application of solvable structures to the reduction of ODEs with a lack of local symmetries is given. Solvable structures considered here are all defined in a nonlocal extension, or covering space, of a given ODE. Examples of the reduction procedure are provided

Ricci flat 4-metrics with bidimensional null orbits : Part I. General Aspects and Nonabelian Case ; Part II. The Abelian Case

Pseudo-Riemannian 4-metrics with bidimensional null Killing orbits are studied. Both Lorentzian and Kleinian (or neutral) cases, are treated simultaneously. Under the assumption that the distribution orthogonal to the orbits is completely integrable a complete exact description of Ricci flat metrics admitting a bidimensional nonabelian Killing algebra is found