1,304 research outputs found
Michel theory of symmetry breaking and gauge theories
We extend Michel's theorem on the geometry of symmetry breaking [L. Michel,
{\it Comptes Rendus Acad. Sci. Paris} {\bf 272-A} (1971), 433-436] to the case
of pure gauge theories, i.e. of gauge-invariant functionals defined on the
space of connections of a principal fiber bundle. Our proof follows
closely the original one by Michel, using several known results on the geometry
of . The result (and proof) is also extended to the case of gauge
theories with matter fields.Comment: 24 pages. An old paper posted for archival purpose
Quaternionic integrable systems
Standard (Arnold-Liouville) integrable systems are intimately related to
complex rotations. One can define a generalization of these, sharing many of
their properties, where complex rotations are replaced by quaternionic ones.
Actually this extension is not limited to the integrable case: one can define a
generalization of Hamilton dynamics based on hyperKahler structures.Comment: 10 pages. To appear in the proceedings of the SPT2002 conference,
edited by S. Abenda, G. Gaeta and S. Walcher, World Scientifi
On a priori energy estimates for characteristic boundary value problems
Motivated by the study of certain non linear free-boundary value problems for hyperbolic systems of partial differential equations arising in Magneto-Hydrodynamics, in this paper we show that an a priori estimate of the solution to certain boundary value problems, in the conormal Sobolev space H1_tan, can be transformed into an L2 a priori estimate of the same problem
The geometry of differential constraints for a class of evolution PDEs
The problem of computing differential constraints for a family of evolution PDEs is discussed from a constructive point of view. A new method, based on the existence of generalized characteristics for evolution vector fields, is proposed in order to obtain explicit differential constraints for PDEs belonging to this family. Several examples, with applications in non-linear stochastic filtering theory, stochastic perturbation of soliton equations and non-isospectral integrable systems, are discussed in detail to verify the effectiveness of the method
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
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
Symmetries of stochastic differential equations using Girsanov transformations
Aiming at enlarging the class of symmetries of an SDE, we introduce a family of stochastic transformations able to change also the underlying probability measure exploiting Girsanov Theorem and we provide new determining equations for the infinitesimal symmetries of the SDE. The well-defined subset of the previous class of measure transformations given by Doob transformations allows us to recover all the Lie point symmetries of the Kolmogorov equation associated with the SDE. This gives the first stochastic interpretation of all the deterministic symmetries of the Kolmogorov equation. The general theory is applied to some relevant stochastic models
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