arXiv subjects

David Iglesias Ponte

Publications and source records attributed to David Iglesias Ponte.

9 recordsLinked to original sources

arXiv2026

Symplectic groupoid multiplication and Poisson integrators

In this paper, we study the role of local symplectic groupoid multiplication in the construction of approximation methods for Hamiltonian flows on the underlying Poisson manifold. We show it can be used to construct a novel multiplicative type of approximating sequences for the relevant Lagrangian bisection. We describe the corresponding theoretical results, their practical implementation into approximation methods on coordinate charts via generating functions, and illustrative examples.

math.SG↗

arXiv2024

Unimodularity and invariant volume forms for Hamiltonian dynamics on coisotropic Poisson homogeneous spaces

In this paper, we introduce the notion of a multiplicative unimodularity for a coisotropic Poisson homogeneous space. Then, we discuss the unimodularity and the multiplicative unimodularity for these spaces and the existence of an invariant volume form for explicit Hamiltonian systems on such spaces. Several interesting examples illustrating the theoretical results are also presented.

math.DG↗

arXiv2023

Cosymplectic groupoids

A cosymplectic groupoid is a Lie groupoid with a multiplicative cosymplectic structure. We provide several structural results for cosymplectic groupoids and we discuss the relationship between cosymplectic groupoids, Poisson groupoids of corank 1, and oversymplectic groupoids of corank 1.

math.SG↗

arXiv2012

Morse families in optimal control problems

We geometrically describe optimal control problems in terms of Morse families in the Hamiltonian framework. These geometric structures allow us to recover the classical first order necessary conditions for optimality and the starting point to run an integrability algorithm. Moreover the integrability algorithm is adapted to optimal control problems in such a way that the trajectories originated by discontinuous controls are also obtained. From the Hamiltonian viewpoint we obtain the equations of motion for optimal control problems in the Lagrangian formalism by means of a proper Lagrangian submanifold. Singular optimal control problems and overdetermined ones are also studied along the paper.

math.OC↗

arXiv2009

Integrability of Poisson-Lie group actions

We establish a 1:1 correspondence between Poisson-Lie group actions on integrable Poisson manifolds and twisted multiplicative hamiltonian actions on source 1-connected symplectic groupoids. For an action of a Poisson-Lie group $G$ on a Poisson manifold $M$, we find an explicit description of the lifted hamiltonian action on the symplectic groupoid $Σ(M)$. We give applications of these results to the integration of Poisson quotients $M/G$, Lu-Weinstein quotients $μ^{-1}(e)/G$ and Poisson homogeneous spaces $G/H$.

math.DG↗

arXiv2008

Hamiltonian spaces for Manin pairs over manifolds

We introduce the notion of Hamiltonian spaces for Manin pairs over manifolds, using the so-called generalized Dirac structures. As an example, we describe Hamiltonian spaces of a quasi-Lie bialgebroid using this general framework. We also discuss reduction of Hamiltonian spaces of this general type.

math.DG↗

arXiv2008

Courant morphisms and moment maps

We study Hamiltonian spaces associated with pairs (E,A), where E is a Courant algebroid and A\subset E is a Dirac structure. These spaces are defined in terms of morphisms of Courant algebroids with suitable compatibility conditions. Several of their properties are discussed, including a reduction procedure. This set-up encompasses familiar moment map theories, such as group-valued moment maps, and it provides an intrinsic approach from which different geometrical descriptions of moment maps can be naturally derived. As an application, we discuss the relationship between quasi-Poisson and presymplectic groupoids.

math.SG↗

arXiv2005

Integration of Dirac-Jacobi structures

We study precontact groupoids whose infinitesimal counterparts are Dirac-Jacobi structures. These geometric objects generalize contact groupoids. We also explain the relationship between precontact groupoids and homogeneous presymplectic groupoids. Finally, we present some examples of precontact groupoids.

math.DG↗

arXiv2005

Universal lifting theorem and quasi-Poisson groupoids

We prove the universal lifting theorem: for an $α$-simply connected and $α$-connected Lie groupoid $\gm$ with Lie algebroid $A$, the graded Lie algebra of multi-differentials on $A$ is isomorphic to that of multiplicative multi-vector fields on $\gm$. As a consequence, we obtain the integration theorem for a quasi-Lie bialgebroid, which generalizes various integration theorems in the literature in special cases. The second goal of the paper is the study of basic properties of quasi-Poisson groupoids. In particular, we prove that a group pair $(D, G)$ associated to a Manin quasi-triple $(\mathfrak d, \mathfrak g, \mathfrak h)$ induces a quasi-Poisson groupoid on the transformation groupoid $G\times D/G\toto D/G$. Its momentum map corresponds exactly with the $D/G$-momentum map of Alekseev and Kosmann-Schwarzbach.

math.DG↗