arXiv subjects

Arnau Planas

Publications and source records attributed to Arnau Planas.

3 recordsLinked to original sources

arXiv2023

Action-angle coordinates and KAM theory for singular symplectic manifolds

This monograph explores classification and perturbation problems for integrable systems on a class of Poisson manifolds called $b^m$-Poisson manifolds. Even if the class of $b^m$-Poisson manifolds is not ample enough to represent general Poisson manifolds, this investigation can be seen as a first step for the study of perturbation theory for general Poisson manifolds. We prove an action-angle coordinate and a KAM theorem for $b^m$-Poisson manifolds which improves the one obtained for $b$-Poisson manifolds for $m=1$ in [KMS16]. As an outcome of this result together with the extension of the desingularization techniques of Guillemin-Miranda-Weitsman to the realm of integrable systems, we obtain a KAM theorem for folded symplectic manifolds. We also obtain a new KAM theorem for symplectic manifolds where the perturbation keeps track of a distinguished hypersurface. In several problems in celestial mechanics, this distinguished hypersurface can be the line at infinity or can represent the collision set.

math.SG↗

arXiv2017

An Invitation to Singular Symplectic Geometry

In this paper we analyze in detail a collection of motivating examples to consider $b^m$-symplectic forms and folded-type symplectic structures. In particular, we provide models in Celestial Mechanics for every $b^m$-symplectic structure. At the end of the paper, we introduce the odd-dimensional analogue to $b$-symplectic manifolds: $b$-contact manifolds.

math.SG↗

arXiv2017

Equivariant classification of $b^m$-symplectic surfaces and Nambu structures

In this paper we extend the classification scheme in [S] for $b^m$-symplectic surfaces and, more generally, $b^m$-Nambu structures to the equivariant setting. When the compact group is the group of deck-transformations of an orientable covering, this yields the classification of these objects for non-orientable surfaces. The paper also includes recipes to construct $b^m$-symplectic structures on surfaces. Feasibility of such constructions depends on orientability and on the colorability of an associated graph. The desingularization technique in [GMW] is revisited for surfaces and the compatibility with this classification scheme is analyzed. We recast the strategy used in [Mt] to classify stable Nambu structures of top degree on orientable manifolds to classify $b^m$-Nambu structures (not necessarily oriented) using the language of $b^m$-cohomology. The paper ends up with an equivariant classification theorem of $b^m$-Nambu structures of top degree.

math.SG↗