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

Swinging Atwood's Machine: Experimental and Theoretical Studies

A Swinging Atwood Machine (SAM) is built and some experimental results concerning its dynamic behaviour are presented. Experiments clearly show that pulleys play a role in the motion of the pendulum, since they can rotate and have non-negligible radii and masses. Equations of motion must therefore take into account the inertial momentum of the pulleys, as well as the winding of the rope around them. Their influence is compared to previous studies. A preliminary discussion of the role of dissipation is included. The theoretical behaviour of the system with pulleys is illustrated numerically, and the relevance of different parameters is highlighted. Finally, the integrability of the dynamic system is studied, the main result being that the Machine with pulleys is non-integrable. The status of the results on integrability of the pulley-less Machine is also recalled.Comment: 37 page

Analysis and design of a drain water heat recovery storage unit based on PCM plates

© 2016. This version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/This paper is focused on the detailed analysis of a PCM plate heat storage unit with a particular configuration, looking for the maximum contact area with the fluid (water) and the minimum volume to be used in a real household application. In that sense, a numerical study of the thermal and fluid dynamic behaviour of the water flow and the PCM melting-solidification processes, together with the thermal behaviour of the solid elements of the unit, has been carried out. On the other hand, an experimental set-up has been designed and built to validate the numerical model and characterise the performance of the heat storage unit. The purpose of the numerical and experimental study is to present a series of results to describe the heat storage unit performance in function of the time. Thus, after a preliminary design study three different cases have been simulated and tested. A 7.2% of discrepancy between numerical results and experimental data has been evaluated for the heat transfer. The PCM heat storage unit designed is capable to store approx. 75% of the thermal energy from the previous process wasted water heat, and recover part of it to supply around 50% of the thermal energy required to heat up the next process.Peer ReviewedPostprint (author's final draft

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

Monitoring subaquatic vegetation using Sentinel-2 imagery in Gallocanta Lake (Aragón, Spain)

Remote sensing allows the study of aquatic vegetation cover in shallow lakes from the different spectral responses of the water as the vegetation grows from the bottom toward the surface. In the case of Gallocanta Lake, its seasonality and shallow depth (less than 2 m) allow us to appreciate the variations in the aquatic vegetation with the apparent color. Six common vegetation indices were tested, and the one with the best response was the so-called NDI45, which uses the normalized ratio between the far red (705 nm) and red (665 nm) bands. Our aims are to show the variations in the surface area covered by vegetation at the bottom of the lagoon, its growth and disappearance when drying occurs, and recolonization in a flooding period. The degree of cover reaches 90% at the most favorable times of the year, generally in summer and coinciding with flooding of the lake. The studied period shows how this method can be used for lacustrine habitat detection and highlights the need for field vegetation inventories in future works, which will allow the spectral measurements to be related to the species present

Algebraic Solutions of the Lam\'e Equation, Revisited

A minor error in the necessary conditions for the algebraic form of the Lam\'e equation to have a finite projective monodromy group, and hence for it to have only algebraic solutions, is pointed out. [See F. Baldassarri, "On algebraic solutions of Lam\'e's differential equation", J. Differential Equations 41 (1981), 44-58.] It is shown that if the group is the octahedral group S_4, then the degree parameter of the equation may differ by +1/6 or -1/6 from an integer; this possibility was missed. The omission affects a recent result on the monodromy of the Weierstrass form of the Lam\'e equation. [See R. C. Churchill, "Two-generator subgroups of SL(2,C) and the hypergeometric, Riemann, and Lam\'e equations", J. Symbolic Computation 28 (1999), 521-545.] The Weierstrass form, which is a differential equation on an elliptic curve, may have, after all, an octahedral projective monodromy group.Comment: 20 pages, elsart document class, no figure

Non integrability of a self-gravitating Riemann liquid ellipsoid

We prove that the motion of a triaxial Riemann ellipsoid of homogeneous liquid without angular momentum does not possess an additional first integral which is meromorphic in position, impulsions, and the elliptic functions which appear in the potential, and thus is not integrable. We prove moreover that this system is not integrable even on a fixed energy level hypersurface.Comment: 14 pages, 8 reference

On reducing the Heun equation to the hypergeometric equation

The reductions of the Heun equation to the hypergeometric equation by polynomial transformations of its independent variable are enumerated and classified. Heun-to-hypergeometric reductions are similar to classical hypergeometric identities, but the conditions for the existence of a reduction involve features of the Heun equation that the hypergeometric equation does not possess; namely, its cross-ratio and accessory parameters. The reductions include quadratic and cubic transformations, which may be performed only if the singular points of the Heun equation form a harmonic or an equianharmonic quadruple, respectively; and several higher-degree transformations. This result corrects and extends a theorem in a previous paper, which found only the quadratic transformations. [See K. Kuiken, "Heun's equation and the hypergeometric equation", SIAM Journal on Mathematical Analysis 10:3 (1979), 655-657.]Comment: 36 pages, a few additional misprints correcte

Seguimiento del fenómeno blanco de la laguna de la Cruz (Cuenca, España)

En el presente estudio se realizó un seguimiento de cinco años por teledetección del fenómeno de la precipitación de carbonato cálcico en la laguna kárstica de la Gitana o de la Cruz (situada en la provincia de Cuenca, España). Se conoce el importante papel que desempeña la precipitación del carbonato cálcico en la ecología del lago ya que influye en las migraciones verticales del fitoplancton, en la concentración de fósforo biodisponible y por ende, en la eutrofización y calidad de las aguas. El blanqueamiento sucede entre los meses de julio y agosto, pudiendo estudiarse en estas fechas a través de sus propiedades ópticas, con el objetivo principal de ofrecer datos actualizados de un fenómeno tradicionalmente estudiado y establecer posibles relaciones entre factores abióticos como la temperatura y/o precipitaciones. Los datos de temperatura del aire, recogidos por la estación meteorológica de Cuenca, sugieren una posible relación entre la aparición del fenómeno blanco y un pulso de temperaturas máximas previas. Por otra parte, no se encontró relación aparente entre las precipitaciones y el blanqueamiento del agua

A direct urea fuel cell - power from fertiliser and waste

For the first time, a working direct urea and direct urine fuel cell has been developed to generate electricity directly from urea or urine