arXiv subjects

J. C. Ruíz-Pantaleón

Publications and source records attributed to J. C. Ruíz-Pantaleón.

8 recordsLinked to original sources

arXiv2025

On the First Cohomology of Infinitesimal Poisson Algebras

For the so-called infinitesimal Poisson algebras encoding first-order jets of Poisson submanifolds, we provide a description of their first cohomology in terms of intrinsic cohomologies of the underlying Poisson submanifold. We establish a natural mapping from their first cohomology to the first Poisson cohomology of the corresponding Poisson submanifold. Moreover, we formulate necessary and sufficient conditions for the vanishing of the first cohomology of infinitesimal Poisson algebras. Finally, we consider the special cases of symplectic leaves and, more generally, Poisson submanifolds with partially split first-order jets. In particular, we derive cohomological necessary conditions for the partially split property.

math.SG↗

arXiv2023

On Gauge Poisson Brackets with Prescribed Symmetry

In the context of averaging method, we describe a reconstruction of invariant connection-dependent Poisson structures from canonical actions of compact Lie groups on fibered phase spaces. Some symmetry properties of Wong's type equations are derived from the main results.

math.DG↗

arXiv2023

Poisson Structures on Trivial Extension Algebras

We present a class of Poisson structures on trivial extension algebras which generalize some known structures induced by Poisson modules. We show that there exists a one-to-one correspondence between such a class of Poisson structures and some data involving (not necessarily flat) contravariant derivatives, and then we give a formulation of this result in terms of Lie algebroids. Some properties of the first Poisson cohomology are also presented. Examples coming from Poisson modules and Poisson submanifolds are given.

math.RA↗

arXiv2022

Areas on the space of smooth probability density functions on $S^2$

We present symbolic and numerical methods for computing Poisson brackets on the spaces of measures with positive densities of the plane, the 2-torus, and the 2-sphere. We apply our methods to compute symplectic areas of finite regions for the case of the 2-sphere, including an explicit example for Gaussian measures with positive densities.

math.SG↗

arXiv2021

On Computational Poisson Geometry II: Numerical Methods

We present twelve numerical methods for evaluation of objects and concepts from Poisson geometry. We describe how each method works with examples, and explain how it is executed in code. These include methods that evaluate Hamiltonian and modular vector fields, compute the image under the coboundary and trace operators, the Lie bracket of differential 1-forms, gauge transformations, and normal forms of Lie-Poisson structures on $\mathbf{R}^{3}$. The complexity of each of our methods is calculated, and we include experimental verifications on examples in dimensions two and three.

math.DG↗

arXiv2020

Infinitesimal Poisson Algebras and Linearization of Hamiltonian Systems

Using the notion of a contravariant derivative, we give some algebraic and geometric characterizations of Poisson algebras associated to the infinitesimal data of Poisson submanifolds. We show that such a class of Poisson algebras provides a suitable framework for the study of the Hamiltonization problem for the linearized dynamics along Poisson submanifolds.

math.DG↗

arXiv2019

On Computational Poisson Geometry I: Symbolic Foundations

We present a computational toolkit for (local) Poisson-Nijenhuis calculus on manifolds. Our python module $\textsf{PoissonGeometry}$ implements our algorithms, and accompanies this paper. We include two examples of how our methods can be used, one for gauge transformations of Poisson bivectors in dimension 3, and a second one that determines parametric Poisson bivector fields in dimension 4.

math.DG↗

arXiv2018

Compatible Poisson Structures on Fibered $5$-Manifolds

We study a class of Poisson tensors on a fibered manifold which are compatible with the fiber bundle structure by the so-called almost coupling condition. In the case of a $5$-dimensional orientable fibered manifolds with $2$-dimensional bases, we describe a global behavior of almost coupling Poisson tensors and their singularities by using a bigraded factorization of the Jacobi identity. In particular, we present some unimodularity criteria and describe a class of gauge type transformations preserving the almost coupling property.

math.SG↗