Related papers: Non-integrability of some Hamiltonians with ration…

The Polchinski equations for the Wilsonian renormalization group in the $D$--dimensional matrix scalar field theory can be written at large $N$ in a Hamiltonian form. The Hamiltonian defines evolution along one extra holographic dimension…

In this article we discuss a weaker version of Liouville's theorem on the integrability of Hamiltonian systems. We show that in the case of Tonelli Hamiltonians the involution hypothesis on the integrals of motion can be completely dropped…

We consider natural Hamiltonian systems of $n>1$ degrees of freedom with polynomial homogeneous potentials of degree $k$. We show that under a genericity assumption, for a fixed $k$, at most only a finite number of such systems is…

We give normal forms for generic k-dimensional parametric families $(Z_\varepsilon)_\varepsilon$ of germs of holomorphic vector fields near $0\in\mathbb{C}^2$ unfolding a saddle-node singularity $Z_0$, under the condition that there exists…

Let $\mathbf{k}$ be a differential field and let $[A]\,:\,Y'=A\,Y$ be a linear differential system where $A\in\mathrm{Mat}(n\,,\,\mathbf{k})$. We say that $A$ is in a reduced form if $A\in\mathfrak{g}(\bar{\mathbf{k}})$ where $\mathfrak{g}$…

In this paper, we investigate solvable structures associated to Hamiltonian equations. For a completely integrable Hamiltonian system with $n$ degrees of freedom, we construct a canonical solvable structure consisting of $2n$ Hamiltonian…

We introduce the concept of Hamiltonian potential variables to map Hamiltonian operators into symplectic operators in a dual space. This generalises the classical trick of switching to a potential variable to obtain a Lagrangian density for…

We integrate with hyperelliptic functions a two-particle Hamiltonian with quartic potential and additionnal linear and nonpolynomial terms in the Liouville integrable cases 1:6:1 and 1:6:8.

We consider non-stationary dynamical systems with one-and-a-half degrees of freedom. We are interested in algorithmic construction of rich classes of Hamilton's equations with the Hamiltonian H=p^2/2+V(x,t) which are Liouville integrable.…

An infinite family of classical superintegrable Hamiltonians defined on the N-dimensional spherical, Euclidean and hyperbolic spaces are shown to have a common set of (2N-3) functionally independent constants of the motion. Among them, two…

We construct solutions of analogues of the nonstationary Schr\"odinger equation corresponding to the polynomial isomonodromic Hamiltonian Garnier system with two degrees of freedom. This solutions are obtained from solutions of systems of…

A universal algorithm to construct N-particle (classical and quantum) completely integrable Hamiltonian systems from representations of coalgebras with Casimir element is presented. In particular, this construction shows that quantum…

Classical integrable Hamiltonian systems generated by elements of the Poisson commuting ring of spectral invariants on rational coadjoint orbits of the loop algebra $\wt{\gr{gl}}^{+*}(2,{\bf R})$ are integrated by separation of variables in…

We discuss some families of integrable and superintegrable systems in $n$-dimensional Euclidean space which are invariant to $m\geq n-2$ rotations. The integrable invariant Hamiltonian $H=\sum p_i^2+V(q)$ commutes with $n-2$ integrals of…

We show, in general, how to transform the nonautonomous nonlinear Schroedinger equation with quadratic Hamiltonians into the standard autonomous form that is completely integrable by the familiar inverse scattering method in nonlinear…

We consider the problem of $n$ points with positive masses interacting pairwise with forces inversely proportional to the distance between them. In particular, it is the classical gravitational, Coulomb or photo-gravitational $n$-body…

We aim to completely formalize the rough topological analysis of integrable Hamiltonian systems admitting analytical solutions such that the initial phase variables along with the time derivatives of the auxiliary variables are expressed as…

The quantum $H_3$ integrable system is a 3D system with rational potential related to the non-crystallographic root system $H_3$. It is shown that the gauge-rotated $H_3$ Hamiltonian as well as one of the integrals, when written in terms of…

The Lie-Poisson algebra so(N+1) and some of its contractions are used to construct a family of superintegrable Hamiltonians on the ND spherical, Euclidean, hyperbolic, Minkowskian and (anti-)de Sitter spaces. We firstly present a…

In this paper, we propose integrable discretizations of a two-dimensional Hamiltonian system with quartic potentials. Using either the method of separation of variables or the method based on bilinear forms, we construct the corresponding…