6 research outputs found
On the geometry of lambda-symmetries, and PDEs reduction
We give a geometrical characterization of -prolongations of vector
fields, and hence of -symmetries of ODEs. This allows an extension to
the case of PDEs and systems of PDEs; in this context the central object is a
horizontal one-form , and we speak of -prolongations of vector fields
and -symmetries of PDEs. We show that these are as good as standard
symmetries in providing symmetry reduction of PDEs and systems, and explicit
invariant solutions
On the relation between standard and -symmetries for PDEs
We give a geometrical interpretation of the notion of -prolongations of
vector fields and of the related concept of -symmetry for partial
differential equations (extending to PDEs the notion of -symmetry for
ODEs). We give in particular a result concerning the relationship between
-symmetries and standard exact symmetries. The notion is also extended to
the case of conditional and partial symmetries, and we analyze the relation
between local -symmetries and nonlocal standard symmetries.Comment: 25 pages, no figures, latex. to be published in J. Phys.
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.