José A. Vallejo

The symplectic structure of Euler–Lagrange superequations and Batalin–Vilkoviski formalism

We study the graded Euler–Lagrange equations from the viewpoint of graded Poincare–Cartan forms. An application to a certain class of solutions of the Batalin–Vilkoviski master equation is also given.

THE POINCARÉ–CARTAN FORM IN SUPERFIELD THEORY

An intrinsic description of the Hamilton–Cartan formalism for first-order Berezinian variational problems determined by a submersion of supermanifolds is given. This is achieved by studying the associated higher-order graded variational problem through the Poincare–Cartan form. Noether theorem and examples from superfield theory and supermechanics are also discussed.

Higher order corrections to adiabatic invariants of generalized slow-fast Hamiltonian systems

We present a coordinate-free approach for constructing approximate first integrals of generalized slow-fast Hamiltonian systems, based on the global averaging method on parameter-dependent phase spaces with S1-symmetry. Explicit global formulas for approximate second-order first integrals are derived. As examples, we analyze the case quadratic in the fast variables (in particular, the elastic pendulum), and the charged particle in a slowly-varying magnetic field.

Nambu-Poisson manifolds and associated n-ary Lie algebroids

We introduce an n-ary Lie algebroid canonically associated with a Nambu– Poisson manifold. We also prove that every Nambu–Poisson bracket defined on functions is induced by some differential operator on the exterior algebra, and characterizes such operators. Some physical examples are presented.

A perturbation theory approach to the stability of the Pais-Uhlenbeck oscillator

We present a detailed analysis of the orbital stability of the Pais-Uhlenbeck oscillator, using Lie-Deprit series and Hamiltonian normal form theories. In particular, we explicitly describe the reduced phase space for this Hamiltonian system and give a proof for the existence of stable orbits for a certain class of self-interaction, found numerically in previous works, by using singular symplectic reduction.

An Approach to Omni-Lie Algebroids Using Quasi-Derivations

We introduce the notion of left (and right) quasi-Loday algebroids and a “universal space” for them, called a left (right) omni-Loday algebroid, in such a way that Lie algebroids, omni-Lie algebras and omni-Loday algebroids are particular substructures.

MODULAR CLASS OF EVEN SYMPLECTIC MANIFOLDS

We provide an intrinsic description of the notion of modular class for an even symplectic manifold and study its properties in this coordinate-free formalism.

Modelling the Landing of a Plane in a Calculus Lab

We exhibit a simple model of a plane landing that involves only basic concepts of differential calculus, so it is suitable for a first-year calculus lab. We use the computer algebra system Maxima and the interactive geometry software GeoGebra to do the computations and graphics.

Collaborative Use of KeTCindy and Free Computer Algebra Systems

In mathematics research and education, harmonious use of high-quality mathematical expressions and mathematically precise artwork is desirable. To produce them, an environment in which several kinds of software (including editing tools, computing tools, and high-quality graphics generators) interact in a collaborative way, is needed. To accomplish this, we have developed Cindy, a plug-in for the popular dynamic geometry software Cinderella, to help mathematicians generate high-quality graphical images. Cindy also enables us to bind the high-performance graphics capability of Cinderella with the powerful computing capability of various computer algebra systems. In this paper, we will show some mathematical artwork to illustrate how the collaborative use of