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Juan-Pablo Ortega

Publications and source records attributed to Juan-Pablo Ortega.

At least 73 records · Page 4Linked to original sources
arXiv2008

Symplectic Group Actions and Covering Spaces

For symplectic group actions which are not Hamiltonian there are two ways to define reduction. Firstly using the cylinder-valued momentum map and secondly lifting the action to any Hamiltonian cover (such as the universal cover), and then performing symplectic reduction in the usual way. We show that provided the action is free and proper, and the Hamiltonian holonomy associated to the action is closed, the natural projection from the latter to the former is a symplectic cover. At the same time we give a classification of all Hamiltonian covers of a given symplectic group action. The main properties of the lifting of a group action to a cover are studied.

math.SG↗
arXiv2008

The stochastic Hamilton-Jacobi equation

We extend some aspects of the Hamilton-Jacobi theory to the category of stochastic Hamiltonian dynamical systems. More specifically, we show that the stochastic action satisfies the Hamilton-Jacobi equation when, as in the classical situation, it is written as a function of the configuration space using a regular Lagrangian submanifold. Additionally, we will use a variation of the Hamilton-Jacobi equation to characterize the generating functions of one-parameter groups of symplectomorphisms that allow to rewrite a given stochastic Hamiltonian system in a form whose solutions are very easy to find; this result recovers in the stochastic context the classical solution method by reduction to the equilibrium of a Hamiltonian system.

math.PR↗
arXiv2008

Superposition rules and stochastic Lie-Scheffers systems

This paper proves a version for stochastic differential equations of the Lie-Scheffers Theorem. This result characterizes the existence of nonlinear superposition rules for the general solution of those equations in terms of the involution properties of the distribution generated by the vector fields that define it. When stated in the particular case of standard deterministic systems, our main theorem improves various aspects of the classical Lie-Scheffers result. We show that the stochastic analog of the classical Lie-Scheffers systems can be reduced to the study of Lie group valued stochastic Lie-Scheffers systems; those systems, as well as those taking values in homogeneous spaces are studied in detail. The developments of the paper are illustrated with several examples.

math.PR↗
arXiv2008

Reduction, reconstruction, and skew-product decomposition of symmetric stochastic differential equations

We present reduction and reconstruction procedures for the solutions of symmetric stochastic differential equations, similar to those available for ordinary differential equations. Additionally, we use the local tangent-normal decomposition, available when the symmetry group is proper, to construct local skew-product splittings in a neighborhood of any point in the open and dense principal orbit type. The general methods introduced in the first part of the paper are then adapted to the Hamiltonian case, which is studied with special care and illustrated with several examples. The Hamiltonian category deserves a separate study since in that situation the presence of symmetries implies in most cases the existence of conservation laws, mathematically described via momentum maps, that should be taken into account in the analysis.

math.PR↗
arXiv2007

Stochastic Hamiltonian dynamical systems

We use the global stochastic analysis tools introduced by P. A. Meyer and L. Schwartz to write down a stochastic generalization of the Hamilton equations on a Poisson manifold that, for exact symplectic manifolds, are characterized by a natural critical action principle similar to the one encountered in classical mechanics. Several features and examples in relation with the solution semimartingales of these equations are presented.

math.PR↗
arXiv2006

Openness and convexity for momentum maps

The purpose of this paper is finding the essential attributes underlying the convexity theorems for momentum maps. It is shown that they are of topological nature; more specifically, we show that convexity follows if the map is open onto its image and has the so called local convexity data property. These conditions are satisfied in all the classical convexity theorems and hence they can, in principle, be obtained as corollaries of a more general theorem that has only these two hypotheses. We also prove a generalization of the "Lokal-global-Prinzip" that only requires the map to be closed and to have a normal topological space as domain, instead of using a properness condition. This allows us to generalize the Flaschka-Ratiu convexity theorem to non-compact manifolds.

math.SG↗
arXiv2006

Metric convexity in the symplectic category

We introduce an extension of the standard Local-to-Global Principle used in the proof of the convexity theorems for the momentum map to handle closed maps that take values in a length metric space. This extension is used to study the convexity properties of the cylinder valued momentum map introduced by Condevaux, Dazord, and Molino and allows us to obtain the most general convexity statement available in the literature for momentum maps associated to a symplectic Lie group action.

math.SG↗
arXiv2005

Symmetry and symplectic reduction

This encyclopedia article briefly reviews without proofs some of the main results in symplectic reduction. The article recalls most the necessary prerequisites to understand the main results, namely, group actions, momentum maps, and coadjoint orbits, among others.

math.SG↗
arXiv2005

Poisson reduction

This encyclopedia article briefly reviews without proofs some of the main results in Poisson reduction. The article recalls most the necessary prerequisites to understand the main results.

math.SG↗
arXiv2005

Cotangent bundle reduction

This encyclopedia article briefly reviews without proofs some of the main results in cotangent bundle reduction. The article recalls most the necessary prerequisites to understand the main results.

math.SG↗
arXiv2005

The reduced spaces of a symplectic Lie group action

There exist three main approaches to reduction associated to canonical Lie group actions on a symplectic manifold, namely, foliation reduction, introduced by Cartan, Marsden-Weinstein reduction, and optimal reduction, introduced by the authors. When the action is free, proper, and admits a momentum map these three approaches coincide. The goal of this paper is to study the general case of a symplectic action that does not admit a momentum map and one needs to use its natural generalization, a cylinder valued momentum map introduced by Condevaux, Dazord, and Molino. In this case it will be shown that the three reduced spaces mentioned above do not coincide, in general. More specifically, the Marsden-Weinstein reduced spaces are not symplectic but Poisson and their symplectic leaves are given by the optimal reduced spaces. Foliation reduction produces a symplectic reduced space whose Poisson quotient by a certain Lie group associated to the group of symmetries of the problem equals the Marsden-Weinstein reduced space. We illustrate these constructions with concrete examples, special emphasis being given to the reduction of a magnetic cotangent bundle of a Lie group in the situation when the magnetic term ensures the non-existence of the momentum map for the lifted action. The precise relation of the cylinder valued momentum map with group valued momentum maps for Abelian Lie groups is also given.

math.SG↗
arXiv2004

Relative normal modes for nonlinear Hamiltonian systems

An estimate on the number of distinct relative periodic orbits around a stable relative equilibrium in a Hamiltonian system with continuous symmetry is given. This result constitutes a generalization to the Hamiltonian symmetric framework of a classical result by Weinstein and Moser on the existence of periodic orbits in the energy levels surrounding a stable equilibrium.The estimate obtained is very precise in the sense that it provides a lower bound for the number of relative periodic orbits at each prescribed energy and momentum values neighboring the stable relative equilibrium in question and with any prefixed (spatiotemporal) isotropy subgroup. Moreover, it is easily computable in particular examples. It is interesting to see how in our result the existence of non trivial relative periodic orbits requires (generic) conditions on the higher order terms of the Taylor expansion of the Hamiltonian function, in contrast with the purely quadratic requirements of the Weinstein--Moser Theorem, which emphasizes the highly non linear character of the relatively periodic dynamical objects.

math.DG↗
arXiv2004

The relation between local and global dual pairs

In this note we clarify the relationship between the local and global definitions of dual pairs in Poisson geometry. It turns out that these are not equivalent. For the passage from local to global one needs a connected fiber hypothesis (this is well known), while the converse requires a dimension condition (which appears not to be known). We also provide examples illustrating the necessity of the extra conditions.

math.SG↗
arXiv2004

Asymptotic and Lyapunov stability of Poisson equilibria

This paper includes results centered around three topics, all of them related with the nonlinear stability of equilibria in Poisson dynamical systems. Firstly, we prove an energy-Casimir type sufficient condition for stability that uses functions that are not necessarily conserved by the flow and that takes into account certain asymptotically stable behavior that may occur in the Poisson category. This method is adapted to Poisson systems obtained via a reduction procedure and we show in examples that the kind of stability that we propose is appropriate when dealing with the stability of the equilibria of some constrained systems. Finally, we discuss two situations in which the use of continuous Casimir functions in stability studies is equivalent to the topological stability methods introduced by Patrick {\it et al.}

math.DS↗
arXiv2004

Bifurcation and forced symmetry breaking in Hamiltonian systems

We consider the phenomenon of forced symmetry breaking in a symmetric Hamiltonian system on a symplectic manifold. In particular we study the persistence of an initial relative equilibrium subjected to this forced symmetry breaking. We see that, under certain nondegeneracy conditions, an estimate can be made on the number of bifurcating relative equilibria.

math.DS↗
arXiv2003

Dynamics on Leibniz manifolds

This paper shows that various relevant dynamical systems can be described as vector fields associated to smooth functions via a bracket that defines what we call a Leibniz structure. We show that gradient flows, some dissipative systems, and nonholonomically constrained simple mechanical systems, among other dynamical behaviors, can be described using this mathematical construction that generalizes the standard Poisson bracket currently used in Hamiltonian mechanics. The symmetries of these systems and the associated reduction procedures are described in detail. A number of examples illustrate the theoretical developments in the paper.

math.DS↗
arXiv2002

Optimal reduction

We generalize various symplectic reduction techniques to the context of the optimal momentum map. Our approach allows the construction of symplectic point and orbit reduced spaces purely within the Poisson category under hypotheses that do not necessarily imply the existence of a momentum map. We construct an orbit reduction procedure for canonical actions on a Poisson manifold that exhibits an interesting interplay with the von Neumann condition previously introduced by the author in his study of singular dual pairs. This condition ensures that the orbits in the momentum space of the optimal momentum map (we call them polar reduced spaces) admit a presymplectic structure that generalizes the Kostant--Kirillov--Souriau symplectic structure of the coadjoint orbits in the dual of a Lie algebra. Using this presymplectic structure, the optimal orbit reduced spaces are symplectic with a form that satisfies a relation identical to the classical one obtained by Marle, Kazhdan, Kostant, and Sternberg for free Hamiltonian actions on a symplectic manifold. In the symplectic case we provide a necessary and sufficient condition for the polar reduced spaces to be symplectic. In general, the presymplectic polar reduced spaces are foliated by symplectic submanifolds that are obtained through a generalization to the optimal context of the so called Sjamaar Principle, already existing in the theory of Hamiltonian singular reduction. We use these ideas in the construction of a family of presymplectic homogeneous manifolds and of its symplectic foliation and we show that these reduction techniques can be implemented in stages in total analogy with the case of free globally Hamiltonian proper actions.

math.SG↗