This paper explains the recent developments on the symplectic theory of Hamiltonian completely integrable systems on symplectic 4-manifolds, compact or not. One fundamental ingredient of these developments has been the understanding of singular affine structures. These developments make use of results obtained by many authors in the second half of the twentieth century, notably Arnold, Duistermaat and Eliasson, of which we also give a concise survey. As a motivation, we present a collection of remarkable results proven in the early and mid 1980s in the theory of Hamiltonian Lie group actions by Atiyah, Guillemin-Sternberg and Delzant among others, and which inspired many people, including the authors, to work on more general Hamiltonian systems. The paper concludes discussing a spectral conjecture for quantum integrable systems.
In the present paper, we consider a smooth $C^\infty$ symplectic classification of Lagrangian fibrations near cusp singularities, parabolic orbits and cuspidal tori. We show that for these singularities as well as for an arrangement of singularities known as a flap, which arises in the integrable subcritical Hamiltonian Hopf bifurcation, the action variables form a complete set of $C^\infty$ symplectic invariants. We also give a symplectic classification for parabolic orbits in the real-analytic case. Namely, we prove that a complete symplectic invariant in this case is given by a real-analyti…
Exponential Stability Estimate of Symplectic Integrators for Integrable Hamiltonian Systems
Inner Mongolia University（内蒙古大学）；Institute of Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学研究所）；South China University of Technology（华南理工大学）
Measure and capacity of wandering domains in Gevrey near-integrable exact symplectic systems
一个泛同态的游荡域是一个开连通集，其迭代是两两不相交的。我们赋予A^n = T^n x R^n通常的精确辛结构。一个可积的单同态Φ^h，即一个只依赖于作用变量的哈密顿量h的时一映射，没有非空游荡域。本文的目的是估计游荡域的大小（测度和Gromov容量）的情况下，一个精确的辛扰动Φ^h，在分析或Gevrey范畴。上估计与Nekhoroshev理论有关，下估计与Arnold扩散的例子有关。这是一个贡献的“定量哈密顿扰动理论“在以前的作品中发起的长期稳定性估计和扩散时间的最优性，我们在这里的重点是离散系统，因为这是自然设置研究游荡域。
This paper is an announcement of a result followed with explanations of some ideas behind. The proofs will appear elsewhere. Our goal is to construct a Hamiltonian perturbation of any completely integrable Hamiltonian system with $2n$ degrees of freedom ($n\geq 2$). The perturbation is $C^{\infty}$ small but the resulting flow has positive metric entropy and it satisfies KAM non-degeneracy conditions. The key point is that positive entropy can be generated in an arbitrarily small tubular neighborhood of one trajectory.
The main purpose of this paper is to prove the smooth local orbital linearization theorem for smooth vector fields which admit a complete set of first integrals near a nondegenerate singular point. The main tools used in the proof of this theorem are the formal orbital linearization theorem for formal integrable vector fields, the blowing-up method, and the Sternberg-Chen isomorphism theorem for formally-equivalent smooth hyperbolic vector fields.
Numerical invariant tori of symplectic integrators for integrable Hamiltonian systems
Inner Mongolia University（内蒙古大学）；Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Hua Loo-Keng Key Laboratory of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院华罗庚数学重点实验室）
The two main results of this paper concern the regularity of the invariant foliation of a C0-integrable symplectic twist diffeomorphisms of the 2-dimensional annulus, namely that $\bullet$ the generating function of such a foliation is C1 ; $\bullet$ the foliation is H{ö}lder with exponent 1/2. We also characterize foliations by graphs that are straightenable via a symplectic homeomorphism and prove that every symplectic homeomorphism that leaves invariant all the leaves of a straightenable foliation has Arnol'd-Liouville coordinates, in which the Dynamics restricted to the leaves is conjugate…
The purpose of this article is to show that a $\mathcal{C}^1$ differential system on $\R^n$ which admits a set of $n-1$ independent $\mathcal{C}^2$ conservation laws defined on an open subset $Ω\subseteq \R^n$, is essentially $\mathcal{C}^1$ equivalent on an open and dense subset of $Ω$, with the linear differential system $u^\prime_1=u_1, \ u^\prime_2=u_2,..., \ u^\prime_n=u_n$. The main results are illustrated in the case of two concrete dynamical systems, namely the three dimensional Lotka-Volterra system, and respectively the Euler equations from the free rigid body dynamics.
