The Poisson equations in the nonholonomic Suslov problem: Integrability, meromorphic and hypergeometric solutions

We consider the problem of integrability of the Poisson equations describing spatial motion of a rigid body in the classical nonholonomic Suslov problem. We obtain necessary conditions for their solutions to be meromorphic and show that under some further restrictions these conditions are also sufficient. The latter lead to a family of explicit meromorphic solutions, which correspond to rather special motions of the body in space. We also give explicit extra polynomial integrals in this case. In the more general case (but under one restriction), the Poisson equations are transformed into a generalized third order hypergeometric equation. A study of its monodromy group allows us also to calculate the "scattering" angle: the angle between the axes of limit permanent rotations of the body in space

Sphere rolling on the surface of a cone

We analyse the motion of a sphere that rolls without slipping on a conical surface having its axis in the direction of the constant gravitational field of the Earth. This nonholonomic system admits a solution in terms of quadratures. We exhibit that the only circular of the system orbit is stable and furthermore show that all its solutions can be found using an analogy with central force problems. We also discuss the case of motion with no gravitational field, that is, of motion on a freely falling cone.Comment: 12 pages, 2 figures, to be published in Eur J Phy

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 bi-integrable natural Hamiltonian systems on the Riemannian manifolds

We introduce the concept of natural Poisson bivectors, which generalizes the Benenti approach to construction of natural integrable systems on the Riemannian manifolds and allows us to consider almost the whole known zoo of integrable systems in framework of bi-hamiltonian geometry.Comment: 24 pages, LaTeX with AMSfonts (some new references were added

Conserved presence of G-quadruplex forming sequences in the Long Terminal Repeat Promoter of Lentiviruses

G-quadruplexes (G4s) are secondary structures of nucleic acids that epigenetically regulate cellular processes. In the human immunodeficiency lentivirus 1 (HIV-1), dynamic G4s are located in the unique viral LTR promoter. Folding of HIV-1 LTR G4s inhibits viral transcription; stabilization by G4 ligands intensifies this effect. Cellular proteins modulate viral transcription by inducing/unfolding LTR G4s. We here expanded our investigation on the presence of LTR G4s to all lentiviruses. G4s in the 5'-LTR U3 region were completely conserved in primate lentiviruses. A G4 was also present in a cattle-infecting lentivirus. All other non-primate lentiviruses displayed hints of less stable G4s. In primate lentiviruses, the possibility to fold into G4s was highly conserved among strains. LTR G4 sequences were very similar among phylogenetically related primate viruses, while they increasingly differed in viruses that diverged early from a common ancestor. A strong correlation between primate lentivirus LTR G4s and Sp1/NF\u3baB binding sites was found. All LTR G4s folded: their complexity was assessed by polymerase stop assay. Our data support a role of the lentiviruses 5'-LTR G4 region as control centre of viral transcription, where folding/unfolding of G4s and multiple recruitment of factors based on both sequence and structure may take place