86 research outputs found

### Invariant Forms and Automorphisms of Locally Homogeneous Multisymplectic Manifolds

It is shown that the geometry of locally homogeneous multisymplectic
manifolds (that is, smooth manifolds equipped with a closed nondegenerate form
of degree > 1, which is locally homogeneous of degree k with respect to a local
Euler field) is characterized by their automorphisms. Thus, locally homogeneous
multisymplectic manifolds extend the family of classical geometries possessing
a similar property: symplectic, volume and contact. The proof of the first
result relies on the characterization of invariant differential forms with
respect to the graded Lie algebra of infinitesimal automorphisms, and on the
study of the local properties of Hamiltonian vector fields on locally
multisymplectic manifolds. In particular it is proved that the group of
multisymplectic diffeomorphisms acts (strongly locally) transitively on the
manifold. It is also shown that the graded Lie algebra of infinitesimal
automorphisms of a locally homogeneous multisymplectic manifold characterizes
their multisymplectic diffeomorphisms.Comment: 25 p.; LaTeX file. The paper has been partially rewritten. Some
terminology has been changed. The proof of some theorems and lemmas have been
revised. The title and the abstract are slightly modified. An appendix is
added. The bibliography is update

### Pseudo-distances on symplectomorphism groups and applications to flux theory

Starting from a given norm on the vector space of exact 1-forms of a compact
symplectic manifold, we produce pseudo-distances on its symplectomorphism group
by generalizing an idea due to Banyaga. We prove that in some cases (which
include Banyaga's construction), their restriction to the Hamiltonian
diffeomorphism group is equivalent to the distance induced by the initial norm
on exact 1-forms. We also define genuine "distances to the Hamiltonian
diffeomorphism group" which we use to derive several consequences, mainly in
terms of flux groups.Comment: 21 pages, no figure; v2. various typos corrected, some references
added. Published in Mathematische Zeitschrif

### The Geometry of Integrable and Superintegrable Systems

The group of automorphisms of the geometry of an integrable system is
considered. The geometrical structure used to obtain it is provided by a normal
form representation of integrable systems that do not depend on any additional
geometrical structure like symplectic, Poisson, etc. Such geometrical structure
provides a generalized toroidal bundle on the carrier space of the system.
Non--canonical diffeomorphisms of such structure generate alternative
Hamiltonian structures for complete integrable Hamiltonian systems. The
energy-period theorem provides the first non--trivial obstruction for the
equivalence of integrable systems

### Slow Diffeomorphisms of a Manifold with Two Dimensions Torus Action

The uniform norm of the differential of the n-th iteration of a
diffeomorphism is called the growth sequence of the diffeomorphism. In this
paper we show that there is no lower universal growth bound for volume
preserving diffeomorphisms on manifolds with an effective two dimensions torus
action by constructing a set of volume-preserving diffeomorphisms with
arbitrarily slow growth.Comment: 12 p

### On magnetic leaf-wise intersections

In this article we introduce the notion of a magnetic leaf-wise intersection
point which is a generalization of the leaf-wise intersection point with
magnetic effects. We also prove the existence of magnetic leaf-wise
intersection points under certain topological assumptions.Comment: 43 page

### Contact complete integrability

Complete integrability in a symplectic setting means the existence of a
Lagrangian foliation leaf-wise preserved by the dynamics. In the paper we
describe complete integrability in a contact set-up as a more subtle structure:
a flag of two foliations, Legendrian and co-Legendrian, and a
holonomy-invariant transverse measure of the former in the latter. This turns
out to be equivalent to the existence of a canonical $\R\ltimes \R^{n-1}$
structure on the leaves of the co-Legendrian foliation. Further, the above
structure implies the existence of $n$ contact fields preserving a special
contact 1-form, thus providing the geometric framework and establishing
equivalence with previously known definitions of contact integrability. We also
show that contact completely integrable systems are solvable in quadratures. We
present an example of contact complete integrability: the billiard system
inside an ellipsoid in pseudo-Euclidean space, restricted to the space of
oriented null geodesics. We describe a surprising acceleration mechanism for
closed light-like billiard trajectories

### Locally continuously perfect groups of homeomorphisms

The notion of a locally continuously perfect group is introduced and studied.
This notion generalizes locally smoothly perfect groups introduced by Haller
and Teichmann. Next, we prove that the path connected identity component of the
group of all homeomorphisms of a manifold is locally continuously perfect. The
case of equivariant homeomorphism group and other examples are also considered.Comment: 14 page

### Imprints of the Quantum World in Classical Mechanics

The imprints left by quantum mechanics in classical (Hamiltonian) mechanics
are much more numerous than is usually believed. We show Using no physical
hypotheses) that the Schroedinger equation for a nonrelativistic system of
spinless particles is a classical equation which is equivalent to Hamilton's
equations.Comment: Paper submitted to Foundations of Physic

### Groups of diffeomorphisms and geometric loops of manifolds over ultra-normed fields

The article is devoted to the investigation of groups of diffeomorphisms and
loops of manifolds over ultra-metric fields of zero and positive
characteristics. Different types of topologies are considered on groups of
loops and diffeomorphisms relative to which they are generalized Lie groups or
topological groups. Among such topologies pairwise incomparable are found as
well. Topological perfectness of the diffeomorphism group relative to certain
topologies is studied. There are proved theorems about projective limit
decompositions of these groups and their compactifications for compact
manifolds. Moreover, an existence of one-parameter local subgroups of
diffeomorphism groups is investigated.Comment: Some corrections excluding misprints in the article were mad

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