Integrability of the constrained rigid body

The integrability theory for the differential equations, which describe the motion of an unconstrained rigid body around a fixed point is well known. When there are constraints the theory of integrability is incomplete. The main objective of this paper is to analyze the integrability of the equations of motion of a constrained rigid body around a fixed point in a force field with potential U(γ)=U(γ ₁,γ ₂,γ ₃). This motion subject to the constraint 〈ν,ω〉=0 with ν is a constant vector is known as the Suslov problem, and when ν=γ is the known Veselova problem, here ω=(ω ₁,ω ₂,ω ₃) is the angular velocity and 〈?,?〉 is the inner product of $\mathbb{R}^{3}$ . We provide the following new integrable cases. (i) The Suslov’s problem is integrable under the assumption that ν is an eigenvector of the inertial tensor I and the potential is such that $$U=-\frac{1}{2I_1I_2}\bigl(I_1\mu^2_1+I_2 \mu^2_2\bigr), $$ where I ₁,I ₂, and I ₃ are the principal moments of inertia of the body, μ ₁ and μ ₂ are solutions of the first-order partial differential equation $$\gamma_3 \biggl(\frac{\partial\mu_1}{\partial\gamma_2}- \frac{\partial\mu_2}{\partial \gamma_1} \biggr)- \gamma_2\frac{\partial \mu_1}{\partial\gamma_3}+\gamma_1\frac{\partial\mu_2}{\partial \gamma_3}=0. $$ (ii) The Veselova problem is integrable for the potential $$U=-\frac{\varPsi^2_1+\varPsi^2_2}{2(I_1\gamma^2_2+I_2\gamma^2_1)}, $$ where Ψ ₁ and Ψ ₂ are the solutions of the first-order partial differential equation where $p=\sqrt{I_{1}I_{2}I_{3} (\frac{\gamma^{2}_{1}}{I_{1}}+\frac{\gamma^{2}_{2}}{I_{2}}+ \frac{\gamma^{2}_{3}}{I_{3}} )}$ . Also it is integrable when the potential U is a solution of the second-order partial differential equation where $\tau_{2}=I_{1}\gamma^{2}_{1}+I_{2}\gamma^{2}_{2}+I_{3}\gamma^{2}_{3}$ and $\tau_{3}=\frac{\gamma^{2}_{1}}{I_{1}}+\frac{\gamma^{2}_{2}}{I_{2}}+ \frac{\gamma^{2}_{3}}{I_{3}}$ . Moreover, we show that these integrable cases contain as a particular case the previous known results.     

Periodic solutions induced by an upright position of small oscillations of a sleeping symmetrical gyrostat

The aim of this paper is to provide sufficient conditions for the existence of periodic solutions emerging from an upright position of small oscillations of a sleeping symmetrical gyrostat with equations of motion being α and β parameters satisfying Δ=α ²?4β>0 and $\beta-\frac{\alpha^{2}}{2}\pm \frac{\alpha \sqrt{\varDelta }}{2}<0$ , ε a small parameter and, F ₁ and F ₂ smooth periodic maps in the variable t in resonance p:q with some of the periodic solutions of the system for ε=0, where p and q are positive integers relatively prime. The main tool used is the averaging theory.     

On the dynamics of the rigid body with a fixed point: periodic orbits and integrability

The aim of the present paper is to study the periodic orbits of a rigid body with a fixed point and quasi-spherical shape under the effect of a Newtonian force field given by different small potentials. For studying these periodic orbits, we shall use averaging theory. Moreover, we provide information on the $\mathcal{C}^{1}$ -integrability of these motions.     

Advances on melamine determination by micellar liquid chromatography: A review

Melamine is a toxic triazine which has been recently proven as a threat to human health. It can be ingested by several methods, being the most important the unethical adulteration of protein-rich foodstuff for economic reasons. This review presents several analytical methods, taken from the literature studies, devoted to the determination of melamine in milk, dietetic supplements, drinking and wastewater, swine kidney, plasma, and urine, using micellar liquid chromatography (MLC). We consider that the control of these samples is crucial to prevent and manage melamine intoxication. This technique has been demonstrated as an excellent tool to determine organic compounds in these matrices. We detail the optimization strategy and the obtained results in the different steps of method development, such as sample pretreatment, chromatographic separation, and validation process. The similarities and differences of the procedures have been described and discussed, as well as their advantages. The main ones were the possibility of direct injection and the efficient chromatographic elution, in spite of the complexity of the samples. Besides, it was found that the MLC procedures were fast, easy-to-handle, inexpensive, eco-friendly, safe, and useful for routine analysis. Therefore, they represent an excellent alternative to reliably evaluate the melamine concentration in that kind of samples.     

Bifurcations from nondegenerate families of periodic solutions in Lipschitz systems

The paper addresses the problem of bifurcation of periodic solutions from a normally nondegenerate family of periodic solutions of ordinary differential equations under perturbations. The approach to solve this problem can be described as transforming (by a Lyapunov–Schmidt reduction) the initial system into one which is in the standard form of averaging, and subsequently applying the averaging principle. This approach encounters a fundamental problem when the perturbation is only Lipschitz (nonsmooth) as we do not longer have smooth Lyapunov–Schmidt projectors. The situation of Lipschitz perturbations has been addressed in the literature lately and the results obtained conclude the existence of the bifurcated branch of periodic solutions. Motivated by recent challenges in control theory, we are interested in the uniqueness problem. We achieve this in the case when the Lipschitz constant of the perturbation obeys a suitable estimate.     

Identification of phototransformation products of sildenafil (Viagra) and its N-demethylated human metabolite under simulated sunlight

Recent publications on pharmaceutical monitoring are increasingly covering the field of illicit drugs and lately the forensic evaluation of designing illegal analogs of lifestyle drugs like the phosphodiesterase type 5 (PDE-5) inhibitors Viagra (sildenafil), Levitra (vardenafil) and Cialis (tadalafil). Recently, the presence of all three erectile dysfunction treatment drugs has been reported in wastewaters at very low concentrations. In the environment, contaminants undergo various physical or chemical processes classified into abiotic (photolysis, hydrolysis) and biotic (biodegradation) reactions. Thus, changes in the chemical structure lead to the formation of new transformation products, which may persist in the environment or be further degraded. This study describes the photolysis of sildenafil (SDF) and its human metabolite N-demethylsildenafil (DM-SDF) under simulated solar radiation (Xenon lamp). Following chromatographic separation of the irradiated samples, eight photoproducts in the SDF samples and six photoproducts for DM-SDF were detected and characterized. The combination of ultra performance liquid chromatography-electrospray ionization-quadrupole time-of-flight-mass spectrometry (UPLC-ESI-QToF-MS), liquid chromatography-atmospheric pressure chemical ionization-triple quadrupole mass spectrometry (LC-APCI-QqQ-MS) and hydrogen/deuterium-exchange experiments allowed to propose plausible chemical structures for the photoproducts, taking into account the characteristic fragmentation patterns and the accurate mass measurements. These mass spectral data provided sound evidence for the susceptibility of the piperazine ring toward photodegradation. A gradual breakdown of this heterocyclic structure gave rise to a series of products, which in part were identical for SDF and DM-SDF. The sulfonic acid, as the formal product of sulfonamide hydrolysis, was identified as key intermediate in the photolysis pathway. In both drug/metabolite molecules, phototransformation processes taking place beyond the sulfonamide group were deemed to be of minor relevance.     

Evolution of yttrium trifluoroacetate during thermal decomposition

A detailed analysis of the thermal decomposition of yttrium trifluoroacetate under different atmospheres is presented. Thermogravimetry, differential thermal analysis, and evolved gas analysis have been used for this in situ analysis. Solid residues at different stages have been characterized by means of X-ray diffraction, elemental analysis, Fourier transform infrared spectroscopy, and scanning electron microscopy. The first decomposition stage (310 °C) is exothermic and involves the complete removal of carbon (organic part) and the formation of yttrium fluoride. This process is characterized by a fast mass loss rate. Afterwards, yttria (Y₂O₃) is formed at 1200 °C through a slow process controlled by the out diffusion of fluorine that involves the formation of yttrium oxyfluoride as an intermediate. The evolution of the mass during the decomposition and the structure of the yttria particles is not affected by the presence of oxygen or water. However, when the oxygen (water) partial pressure is as low as 0.02% (<0.002%), the kinetics and final particle structure are strongly affected.