855 research outputs found
A variational principle for volume-preserving dynamics
We provide a variational description of any Liouville (i.e. volume
preserving) autonomous vector fields on a smooth manifold. This is obtained via
a ``maximal degree'' variational principle; critical sections for this are
integral manifolds for the Liouville vector field. We work in coordinates and
provide explicit formulae
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
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
Reduction and reconstruction of stochastic differential equations via symmetries
An algorithmic method to exploit a general class of infinitesimal symmetries
for reducing stochastic differential equations is presented and a natural
definition of reconstruction, inspired by the classical reconstruction by
quadratures, is proposed. As a side result the well-known solution formula for
linear one-dimensional stochastic differential equations is obtained within
this symmetry approach. The complete procedure is applied to several examples
with both theoretical and applied relevance
A priori estimates for 3D incompressible current-vortex sheets
We consider the free boundary problem for current-vortex sheets in ideal
incompressible magneto-hydrodynamics. It is known that current-vortex sheets
may be at most weakly (neutrally) stable due to the existence of surface waves
solutions to the linearized equations. The existence of such waves may yield a
loss of derivatives in the energy estimate of the solution with respect to the
source terms. However, under a suitable stability condition satisfied at each
point of the initial discontinuity and a flatness condition on the initial
front, we prove an a priori estimate in Sobolev spaces for smooth solutions
with no loss of derivatives. The result of this paper gives some hope for
proving the local existence of smooth current-vortex sheets without resorting
to a Nash-Moser iteration. Such result would be a rigorous confirmation of the
stabilizing effect of the magnetic field on Kelvin-Helmholtz instabilities,
which is well known in astrophysics
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.
Stratification of the orbit space in gauge theories. The role of nongeneric strata
Gauge theory is a theory with constraints and, for that reason, the space of
physical states is not a manifold but a stratified space (orbifold) with
singularities. The classification of strata for smooth (and generalized)
connections is reviewed as well as the formulation of the physical space as the
zero set of a momentum map. Several important features of nongeneric strata are
discussed and new results are presented suggesting an important role for these
strata as concentrators of the measure in ground state functionals and as a
source of multiple structures in low-lying excitations.Comment: 22 pages Latex, 1 figur
Strangeness enhancements at central rapidity in 40 A GeV/c Pb-Pb collisions
Results are presented on neutral kaon, hyperon and antihyperon production in
Pb-Pb and p-Be interactions at 40 GeV/c per nucleon. The enhancement pattern
follows the same hierarchy as seen in the higher energy data - the enhancement
increases with the strangeness content of the hyperons and with the centrality
of collision. The centrality dependence of the Pb-Pb yields and enhancements is
steeper at 40 than at 158 A GeV/c. The energy dependence of strangeness
enhancements at mid-rapidity is discussed.Comment: 15 pages, 10 figures and 3 tables. Presented at International
Conference on Strangeness in Quark Matter (SQM2009), Buzios, Rio de Janeiro,
Brazil, 27 Sept - 2 Oct 2009. Submitted to J.Phys.G: Nucl.Part.Phys, one
reference adde
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.
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