Analytic Non-integrability in String Theory

Using analytic techniques developed for Hamiltonian dynamical systems we show that a certain classical string configurations in AdS_5 x X_5 with X_5 in a large class of Einstein spaces, is non-integrable. This answers the question of integrability of string on such backgrounds in the negative. We consider a string localized in the center of AdS_5 that winds around two circles in the manifold X_5.Comment: 14 page

A list of all integrable 2D homogeneous polynomial potentials with a polynomial integral of order at most 4 in the momenta

We searched integrable 2D homogeneous polynomial potential with a polynomial first integral by using the so-called direct method of searching for first integrals. We proved that there exist no polynomial first integrals which are genuinely cubic or quartic in the momenta if the degree of homogeneous polynomial potentials is greater than 4.Comment: 22 pages, no figures, to appear in J. Phys. A: Math. Ge

Darboux points and integrability of homogeneous Hamiltonian systems with three and more degrees of freedom

We consider natural complex Hamiltonian systems with $n$ degrees of freedom given by a Hamiltonian function which is a sum of the standard kinetic energy and a homogeneous polynomial potential $V$ of degree $k>2$. The well known Morales-Ramis theorem gives the strongest known necessary conditions for the Liouville integrability of such systems. It states that for each $k$ there exists an explicitly known infinite set \scM_k\subset\Q such that if the system is integrable, then all eigenvalues of the Hessian matrix V''(\vd) calculated at a non-zero \vd\in\C^n satisfying V'(\vd)=\vd, belong to \scM_k. The aim of this paper is, among others, to sharpen this result. Under certain genericity assumption concerning $V$ we prove the following fact. For each $k$ and $n$ there exists a finite set \scI_{n,k}\subset\scM_k such that if the system is integrable, then all eigenvalues of the Hessian matrix V''(\vd) belong to \scI_{n,k}. We give an algorithm which allows to find sets \scI_{n,k}. We applied this results for the case $n=k=3$ and we found all integrable potentials satisfying the genericity assumption. Among them several are new and they are integrable in a highly non-trivial way. We found three potentials for which the additional first integrals are of degree 4 and 6 with respect to the momenta.Comment: 54 pages, 1 figur

On algebraic construction of certain integrable and super-integrable systems

We propose a new construction of two-dimensional natural bi-Hamiltonian systems associated with a very simple Lie algebra. The presented construction allows us to distinguish three families of super-integrable monomial potentials for which one additional first integral is quadratic, and the second one can be of arbitrarily high degree with respect to the momenta. Many integrable systems with additional integrals of degree greater than two in momenta are given. Moreover, an example of a super-integrable system with first integrals of degree two, four and six in the momenta is found.Comment: 37 page

Resolution of First- and Second-Order Linear Differential Equations with Periodic Inputs by a Computer Algebra System

In signal processing, a pulse means a rapid change in the amplitude of a signal from a baseline value to a higher or lower value, followed by a rapid return to the baseline value. A square wave function may be viewed as a pulse that repeats its occurrence periodically but the return to the baseline value takes some time to happen. When these periodic functions act as inputs in dynamic systems, the standard tool commonly used to solve the associated initial value problem (IVP) is Laplace transform and its inverse. We show how a computer algebra system may also provide the solution of these IVP straight forwardly by adequately introducing the periodic input

Early Cretaceous absolute geomagnetic paleointensities from Córdoba province (Argentina)

We present here new paleointensity and geochronology results from Early Cretaceous volcanic rocks of Sierra Chica de Cordoba (Argentina). The new K-Ar isotopic ages of 5 samples range from 136 to 122 Ma. Twenty five samples from 7 individual flows yielded acceptable paleointensity estimates. The mean paleointensity values per flow are ranging from 53.0±1.9 to 25.4±2.6 μT and the corresponding Virtual Dipole Moments (VDMs) are ranging from 9.3±1.3 to 4.6±0.5 (1022 Am2). This corresponds to the mean value of 7.3±1.7x1022 Am2, which is compatible to the present geomagnetic axial dipole. Currently available selected paleointensity data from 80 to 130 Ma suggest that geomagnetic field strength frequently fluctuated before and during the Cretaceous Normal Superchron while the magnetic polarity maintained stable. The mean paleointensities derived from Cordoba lavas agree remarkably well with those obtained from the Parana Magmatic Province (133-132 Ma). This reinforces the hypothesis about the unreliability of ‘Mesozoic Dipole Low'.Fil: Cejudo Ruiz, Ruben. Universidad Nacional Autónoma de México; MéxicoFil: Goguitchaichvili, Avto. Universidad Nacional Autónoma de México; MéxicoFil: Geuna, Silvana Evangelina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Geociencias Básicas, Aplicadas y Ambientales de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Geociencias Básicas, Aplicadas y Ambientales de Buenos Aires; ArgentinaFil: Alva-Valdivia, Luis M.. Universidad Nacional Autónoma de México; MéxicoFil: Solé, Jesus. Universidad Nacional Autónoma de México; MéxicoFil: Morales, Juan. Universidad Nacional Autónoma de México; Méxic