The Liouville-Arnold-Nekhoroshev theorem for non-compact invariant manifolds

齐次辛哈密顿系统与切触哈密顿系统的Liouville-Arnold定理

Liouville-Arnold theorem for homogeneous symplectic and contact Hamiltonian systems

该研究先推导恰当辛流形上齐次函数的Liouville-Arnold定理，再经辛化证明切触哈密顿系统的对应定理，构造出可线性化运动方程的余迷向子流形叶状结构与作用-角坐标。

Weak Liouville-Arnold Theorems & Their Implications

This paper studies the existence of invariant smooth Lagrangian graphs for Tonelli Hamiltonian systems with symmetries. In particular, we consider Tonelli Hamiltonians with n independent but not necessarily involutive constants of motion and obtain two theorems reminiscent of the Liouville-Arnold theorem. Moreover, we also obtain results on the structure of the configuration spaces of such systems that are reminiscent of results on the configuration space of completely integrable Tonelli Hamiltonians.

Non-integrability of the Painlevé IV equation in the Liouville-Arnold sense and Stokes phenomena

In this paper we study the integrability of the Hamiltonian system associated to the fourth Painlevé equation. We prove that one two parametric family of this Hamiltonian system is not integrable in the sense of the Liouville-Arnold theorem. Computing explicitly the Stokes matrices and the formal invariants of the second variational equations we deduce that the connected component of the unit element of the corresponding Galois grou is not Abelian. Thus the Morales-Ramis-Simó theory leads to a non-integrable result. Moreover, combining the new result with our previous one we establish that for…

Liouville-Arnold integrability for scattering under cone potentials

The problem of scattering of particles on the line with repulsive interactions, gives rise to some well-known integrable Hamiltonian systems, for example, the nonperiodic Toda lattice or Calogero's system. The aim of this note is to outline our researches which proved the integrability of a much larger class of systems, including some that had never been considered, such as the scattering with very-long-range interaction potential. The integrability of all these systems survives any small enough perturbation of the potential in an arbitrary compact set. Our framework is based on the concept of…

Liouville-Arnold integrability of the pentagram map on closed polygons

The pentagram map is a discrete dynamical system defined on the moduli space of polygons in the projective plane. This map has recently attracted a considerable interest, mostly because its connection to a number of different domains, such as: classical projective geometry, algebraic combinatorics, moduli spaces, cluster algebras and integrable systems. Integrability of the pentagram map was conjectured by R. Schwartz and later proved by V. Ovsienko, R. Schwartz and S. Tabachnikov for a larger space of twisted polygons. In this paper, we prove the initial conjecture that the pentagram map is c…

Cotangent models for integrable systems

We associate cotangent models to a neighbourhood of a Liouville torus in symplectic and Poisson manifolds focusing on a special class called $b$-Poisson/$b$-symplectic manifolds. The semilocal equivalence with such models uses the corresponding action-angle coordinate theorems in these settings: the theorem of Liouville-Mineur-Arnold [A74] for symplectic manifolds and an action-angle theorem for regular Liouville tori in Poisson manifolds [LMV11]. Our models comprise regular Liouville tori of Poisson manifolds but also consider the Liouville tori on the singular locus of a $b$-Poisson manifold…

哈密顿系统的可解结构

Solvable Structures for Hamiltonian Systems

本文研究哈密顿方程的可解结构，为完全可积哈密顿系统构造由2n个哈密顿向量场组成的规范可解结构，导出Pfaffian形式并给出作用-角变量，从而以可解结构新解释Arnold--Liouville定理。

Pseudo-integrable billiards and arithmetic dynamics

We introduce a new class of billiard systems in the plane, with boundaries formed by finitely many arcs of confocal conics such that they contain some reflex angles. Fundamental dynamical, topological, geometric, and arithmetic properties of such billiards are studied. The novelty, caused by reflex angles on boundary, induces invariant leaves of higher genera and dynamical behaviour different from Liouville-Arnold's theorem. Its analogue is derived from the Maier theorem on measured foliations. A local version of Poncelet theorem is formulated and necessary algebro-geometric conditions for per…

A conceptual approach to the problem of action-angle variables

In this paper we develop a general conceptual approach to the problem of existence of action-angle variables for dynamical systems, which establishes and uses the fundamental conservation property of associated torus actions: anything which is preserved by the system is also preserved by the associated torus actions. This approach allows us to obtain, among other things: a) the shortest and most conceptual easy to understand proof of the classical Arnold--Liouville--Mineur theorem; b) basically all known results in the literature about the existence of action-angle variables in various context…

Action-angle Variables for Generic 1D Mechanical Systems

We consider a 1D mechanical system $$\bar {\mathtt H}(\mathtt P,\mathtt Q)=\mathtt P^2+\bar {\mathtt G}(\mathtt Q)$$ in action-angle variable $(\mathtt P,\mathtt Q)$ where $\bar {\mathtt G}$ is a $2π$-periodic analytic function with non degenerate critical points. Then, we consider a small analytic perturbation of $\bar {\mathtt H}$ of the form $${\mathtt H}^*(\mathtt P,\mathtt Q;\hat{\mathtt P}) = \mathtt P^2+\bar {\mathtt G}(\mathtt Q)+ η{\mathtt F} (\mathtt P,\mathtt Q;\hat{\mathtt P})=:\mathtt P^2 + {\mathtt G}^*(\mathtt P,\mathtt Q;\hat{\mathtt P})\,, \qquad η\ll 1\ ,$$ where the perturbe…