Gabriel Paternain, Ralf J. Spatzier
We construct Riemannian manifolds with completely integrable geodesic flows, in particular various nonhomogeneous examples. The methods employed are a modification of Thimm's method, Riemannian submersions and connected sums.
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We present a geometric interpretation of integrability of geodesic flow by quadratic integrals in terms of the web theory and construct integrable billiards on surfaces admitting such integrals.
In this survey article we gather classical as well as recent results on minimal geodesics of Riemannian or Finsler metrics, giving special attention to the two-dimensional case. Moreover, we present open problems together with some first ideas as to the solutions.
We propose a new condition $\aleph$ which enables to get new results on integrable geodesic flows on closed surfaces. This paper has two parts. In the first, we strengthen Kozlov's theorem on non-integrability on surfaces of higher genus. In the second, we study integrable geodesic flows on 2-torus. Our main result for 2-torus describes the phase portraits of integrable flows. We prove that they are essentially standard outside, what we call, separatrix chains. The complement to the union of the separatrix chains is $C^0$-foliated by invariant sections of the bundle.
For any toric automorphism with only real eigenvalues a Riemannian metric with an integrable geodesic flow on the suspension of this automorphism is constructed. A qualitative analysis of such a flow on a three-solvmanifold constructed by the authors in math.DG/9905078 is done. This flow is an example of the geodesic flow, which has vanishing Liouville entropy and, moreover, is integrable but has positive topological entropy. The authors also discuss some open problems on integrability of geodesic flows and related subjects.
Integrable sub-Riemannian geodesic flows on the special orthogonal group
本文研究特殊正交群上橡胶滚动分布诱导的次黎曼几何，通过构造Lax对和双哈密顿结构，证明了任意右不变次黎曼度量的几乎所有正规测地线完全可积。
By studying completely integrable torus actions on contact manifolds we prove a conjecture of Toth and Zelditch that toric integrable geodesic flows on tori must have flat metrics.
Finite-gap potentials and integrable geodesic equations on a 2-surface
本文证明一维薛定谔方程可视为二维曲面上的测地线方程，并利用Baker-Akhiezer函数显式构造了有限间隙势情形的度量与测地线。
We explicitely construct an example of an analytic metric on $T^2$ which is non-separable but it is locally integrable on an energy surface. The construction is based on a KAM-like approach and a careful control on what happens on the energy surface.
In this paper the magnetic geodesic flow on a 2-torus is considered. We study a semi-hamiltonian quasi-linear PDEs which is equivalent to the existence of polynomial in momenta first integral of magnetic geodesic flow on fixed energy level. It is known that diagonal metric associated with this system is Egorov one if degree of the first integral is equal to 2 or 3. In this paper we prove this fact in the case of existence of the first integral of any degree.
We completely integrate the magnetic geodesic flow on a flat two-torus with the magnetic field $F = \cos (x) dx \wedge dy$ and describe all contractible periodic magnetic geodesics. It is shown that there are no such geodesics for energy $E \geq 1/2$, for $E< 1/2$ simple periodic magnetic geodesics form two $S^1$-families for which the (fixed energy) action functional is positive and therefore there are no periodic magnetic geodesics for which the action functional is negative.
Integrable geodesic flows with simultaneously diagonalisable quadratic integrals
研究显示，若n维流形上测地流的n个功能独立二次动量积分在每一点的切空间同时对角化，则这些积分源自Stäckel构造，表明度量允许正交分离变量。
This paper is a review of recent and classical results on integrable geodesic flows on Riemannian manifolds and topological obstructions to integrability. We also discuss some open problems.
Motivated by a paper of Bolsinov and Taimanov DG/9911193 we consider non-holonomic situation and exhibit examples of sub-Riemannian metrics with integrable geodesic flows and positive topological entropy. Moreover the Riemannian examples are obtained as "holonomization" of sub-Riemannian ones. A feature of non-holonomic situation is non-compactness of the phase space. We also exhibit a Liouvulle-integrable Hamiltonian system with topological entropy of all integrals positive.
This paper is a review of recent results on integrable nonholonomic geodesic flows of left--invariant metrics and left- and right--invariant constraint distributions on compact Lie groups.
As we have proved in [L], the geodesic flows associated with the flat metrics on T^2 minimize the polynomial entropy. In this paper, we show that, among the geodesic flows that are Bott integrable and dynamically coherent, the geodesic flows associated to flat metrics are local strict minima for the polynomial entropy. To this aim, we prove a graph property for invariant Lagrangian tori in near-integrable systems.
We study the magnetic geodesic flows on 2-surfaces having an additional first integral which is independent of the Hamiltonian at a fixed energy level. The following two cases are considered: when there exists a quadratic in momenta integral, and also the case of a rational in momenta integral with a linear numerator and denominator. In both cases certain semi-Hamiltonian systems of PDEs appear. In this paper we construct exact solutions (generally speaking, local ones) to these systems: in the first case via the generalized hodograph method, in the second case via the Legendre transformation …
Uniformity of geodesic flow in non-integrable 3-manifolds
本文研究非可积有限多立方体平移三维流形上测地流的均匀分布问题，提出分裂方法处理含一个可积方向的笛卡尔积情形，并结合分裂覆盖论证将结果推广到三个方向均不可积且满足特殊条件的流形。
We discuss canonical transformations relating well-known geodesic flows on the cotangent bundle of the sphere with a set of geodesic flows with quartic invariants. By adding various potentials to the corresponding geodesic Hamiltonians, we can construct new integrable systems on the sphere with quartic invariants.
Suppose we are given a compact Riemannian manifold (Q,g)with completely integrable geodesic flow. Let G be a compact connected Lie group acting freely on Q by isometries. The natural question arises: will the geodesic flow on Q/G equipped with the submersion metric be integrable? Under one natural assumption, we prove that the answer is affirmative. New examples of manifolds with completely integrable geodesic flows are obtained.
Carnot groups are subRiemannian manifolds. As such they admit geodesic flows, which are left-invariant Hamiltonian flows on their cotangent bundles. Some of these flows are integrable. Some are not. The space of k-jets for real-valued functions on the real line forms a Carnot group of dimension $k+2$. We show that its geodesic flow is integrable and that its geodesics generalize Euler's elastica, with the case $k=2$ corresponding to the elastica, as shown by Sachkov and Ardentov.
We show that Jacobi fields of a completely integrable Hamiltonian system of m degrees of freedom also make up a completely integrable system. They provide m additional first integrals which characterize a relative motion.
Local high-degree polynomial integrals of geodesic flows and the generalized hodograph method
针对二维曲面测地流的动量高次多项式第一积分问题，将其转化为半 Hamilton 拟线性偏微分方程组，并利用广义速端曲线法构造了次数3、4、5的局部可积示例。
We classify the completely integrable systems associated with classical root systems whose potential functions are meromorphic at an infinite point.
Let (Σ, g) be a compact $C^2$ finslerian 3-manifold. If the geodesic flow of g is completely integrable, and the singular set is a tamely-embedded polyhedron, then $π_1(Σ)$ is almost polycyclic. On the other hand, if Σ is a compact, irreducible 3-manifold and $π_1(Σ)$ is infinite polycyclic while $π_2(Σ)$ is trivial, then Σ admits an analytic riemannian metric whose geodesic flow is completely integrable and singular set is a real-analytic variety. Additional results in higher dimensions are proven.
Time-quantitative density of non-integrable systems
本文提出新方法证明平坦动力系统中测地线的时间定量稠密性，给出斜率劣逼近数时超稠密性的简短新证明，并推广到代数多矩形曲面。
The only one example has been known of magnetic geodesic flow on the 2-torus which has a polynomial in momenta integral independent of the Hamiltonian. In this example the integral is linear in momenta and corresponds to a one parametric group preserving the Lagrangian function of the magnetic flow. In this paper the problem of integrability on one energy level is considered. This problem can be reduced to a remarkable Semi-hamiltonian system of quasi-linear PDEs and to the question of existence of smooth periodic solutions for this system. Our main result states that the pair of Liouville met…
A trivial bundle of regular connected invariant manifolds of a completely integrable Hamiltonian system can be provided with action-angle coordinates.
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.
A time-dependent completely integrable Hamiltonian system is proved to admit the action-angle coordinates around any regular instantly compact invariant manifold. Written relative to these coordinates, its Hamiltonian and first integrals are functions only of action coordinates.
The main purpose of this article is to study from the geometric point of view the problem of limit cycles bifurcation of perturbed completely integrable systems.
We state two recent results concerning the linearization of integrable systems on generalised Jacobians. Then we apply this to the (complexified) spherical pendulum.
Analytic Liouville Integrable Geodesic Flow Has Zero Entropy
本文研究实解析刘维尔可积哈密顿系统的拓扑熵，通过有限正则展开与齐格林主导项论证的遍历版本，证明满足条件的哈密顿流及闭实解析黎曼流形上的可积测地流具有零拓扑熵，肯定了帕特纳因1992年的猜想。
The obstruction to the existence of global action-angle coordinates of Abelian and noncommutative (non-Abelian) completely integrable systems with compact invariant submanifolds has been studied. We extend this analysis to the case of noncompact invariant submanifolds.
Given a completely integrable system, we associate to any connection on its invariant tori fibred over a parameter manifold the classical and quantum holonomy operator (generalized Berry's phase factor), without any adiabatic approximation.
In this paper it is shown that the existence of two independent holomorphic first integrals for foliations by curves on (C^3,0) is not a topological invariant. More precisely, we provide an example of two topologically equivalent foliations such that only one of them admits two independent holomorphic first integrals. The existence of invariant surfaces over which the induced foliation possesses infinitely many separatrices possibly constitutes the sole obstruction for the topological invariance of complete integrability and a characterization of foliations admitting this type of invariant sur…
Melnikov Method for Perturbed Completely Integrable Systems
针对任意维完全可积系统的周期轨道在非线性周期扰动下的保持与稳定性问题，结合Melnikov方法与Floquet理论建立判定条件，并在广义Euler系统与退化谐振子中验证应用。
The goal of this survey is to give a list of resent results about topology of manifolds admitting different metrics with the same geodesics. We emphasize the role of the theory of integrable systems in obtaining these results.
Geometry of Integrable Linkages
本文类比牵引车-挂车问题，定义带无滑动约束的2连杆问题，利用次黎曼几何求测地线，证明其可积并与平面细丝方程建立双重联系。
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 b…
We provide a constructive method designed in order to control the stability of a given periodic orbit of a general completely integrable system. The method consists of a specific type of perturbation, such that the resulting perturbed system becomes a codimension-one dissipative dynamical system which also admits that orbit as a periodic orbit, but whose stability can be a-priori prescribed. The main results are illustrated in the case of a three dimensional dissipative perturbation of the harmonic oscillator, and respectively Euler's equations form the free rigid body dynamics.
We provide a criterion in order to decide the stability of non-degenerate equilibrium states of completely integrable systems. More precisely, given a Hamilton-Poisson realization of a completely integrable system generated by a smooth $n-$ dimensional vector field, $X$, and a non-degenerate regular (in the Poisson sense) equilibrium state, $\overline{x}_e$, we define a scalar quantity, $\mathcal{I}_{X}(\overline{x}_e)$, whose sign determines the stability of the equilibrium. Moreover, if $\mathcal{I}_{X}(\overline{x}_e)>0$, then around $\overline{x}_e$ there exist one-parameter families of pe…
We introduce a sl_2-invariant family of nonlinear vector fields with a non-semisimple triple zero singularity. In this paper we are concerned with characterization and normal form classification of these vector fields. We show that the family constitutes a Lie algebra structure and each vector field from this family is solenoidal, completely integrable and rotational. All such vector fields share a common quadratic invariant. We provide a Poisson structure for the Lie algebra from which the second invariant for each vector field can be readily derived. We show that each vector field from this …
The aim of this paper is to introduce a class of Hamiltonian autonomous systems in dimension 4 which are completely integrable and their dynamics is described in all details. They have an equilibrium point which is stable for some rare elements of the class, and unstable in most cases. Anyhow, it is linearly stable (all orbits of the linearized system are bounded) and no motion is asymptotic in the past, namely no non-constant solution has the equilibrium as limit point as time goes to minus infinity. In the unstable cases, there is a sequence of initial data which converges to the equilibrium…
Given any symplectomorphism on $D^{2n} (n\geq 1)$ which is $C^{\infty}$ close to the identity, and any completely integrable Hamiltonian system $Φ^t_H$ in the proper dimension, we construct a $C^{\infty}$ perturbation of $H$ such that the resulting Hamiltonian flow contains a "local Poincaré section" that "realizes" the symplectomorphism. As a (motivating) application, we show that there are arbitrarily small perturbations of any completely integrable Hamiltonian system which are entropy non-expansive (and, in particular, exhibit hyperbolic behavior on a set of positive measure).
In this paper we construct multiparametric families of two dimensional metrics with polynomial first integral. Such integrable geodesic flows are described by solutions of some semi-Hamiltonian hydrodynamic type system. We find infinitely many conservation laws and commuting flows for this system. This procedure allows us to present infinitely many particular metrics by the generalized hodograph method.
We describe a class of completely integrable $G$-invariant magnetic geodesic flows on (co)adjoint orbits of a compact connected Lie group $G$ with magnetic field given by the Kirillov-Konstant 2-form.
Completely integrable Hamiltonian systems look promising for controllability since their first integrals are stable under an internal evolution, and one may hope to find a perturbation of a Hamiltonian which drives the first integrals at will. Action-angle coordinates are most convenient for this purpose. Written with respect to these coordinates, a Hamiltonian and first integrals depend only on the action ones. We introduce a suitable perturbation of a Hamiltonian by a term containing time-dependent parameters so that an evolution of action variables is nothing else than a displacement along …
Completely Integrable Replicator Dynamics Associated to Competitive Networks
本文构造并证明一族无限多个 Liouville–Arnold 可积的竞争网络复制者方程，据此分类低维锦标赛复制者，并揭示 Allesina–Levine 模型图示具有拟周期动力学。
In this paper we study some aspects of integrable magnetic systems on the two-torus. On the one hand, we construct the first non-trivial examples with the property that all magnetic geodesics with unit speed are closed. On the other hand, we show that those integrable magnetic systems admitting a global surface of section satisfy a sharp systolic inequality.
We study integrals of completely integrable quantum systems associated with classical root systems. We review integrals of the systems invariant under the corresponding Weyl group and as their limits we construct enough integrals of the non-invariant systems, which include systems whose complete integrability will be first established in this paper. We also present a conjecture claiming that the quantum systems with enough integrals given in this note coincide with the systems that have the integrals with constant principal symbols corresponding to the homogeneous generators of the $B_n$-invar…
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.
Let $(M,Ω)$ be a connected symplectic 4-manifold and let $F=(J,H) : M \to \mathbb{R}^2$ be a completely integrable system on $M$ with only non-degenerate singularities and for which $J : M \to \mathbb{R}$ is a proper map. Assume that $F$ does not have singularities with hyperbolic blocks and that $p_1,...,p_n$ are the focus-focus singularities of $F$. For each subset $S=\{i_1,...,i_j\}$ we will show how to modify $F$ locally around any $p_i, i \in S$, in order to create a new integrable system $\tilde{F}=(J, \tilde{H}) : M \to \mathbb{R}^2$ such that its classical spectrum $\tilde{F}(M)$ conta…
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