28 research outputs found
Nonlocal interpretation of -variational symmetry-reduction method
In this paper we give a geometric interpretation of a reduction method based
on the so called -variational symmetry (C. Muriel, J.L. Romero and P.
Olver 2006 \emph{Variational -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 -symmetries and ODEs reduction
A reduction method of ODEs not possessing Lie point symmetries makes use of
the so called -symmetries (C. Muriel and J. L. Romero, \emph{IMA J.
Appl. Math.} \textbf{66}, 111-125, 2001). The notion of covering for an ODE
is used here to recover -symmetries of as
nonlocal symmetries. In this framework, by embedding into a
suitable system determined by the function ,
any -symmetry of can be recovered by a local symmetry of
. As a consequence, the reduction method of Muriel and
Romero follows from the standard method of reduction by differential invariants
applied to .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 -symmetries. These symmetries are not standard
symmetries, nevertheless for any given -symmetry of an ODE one can always reconstruct nonlocal symmetries of %
. As a consequence, using these nonlocal symmetries, the -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