论文标题
凸汉密尔顿系统的轨道段附近的正常形式
Normal form near orbit segments of convex Hamiltonian systems
论文作者
论文摘要
在研究汉密尔顿系统在cotangent束上的研究中，通过添加潜力（仅取决于基点的功能）来扰动Hamiltoni-ans是很自然的。这导致了ma {ñ} {é}的定义：如果给定Hamiltonian H，则属性是通用的，即H + U所满足该属性的一组电位u。这个概念主要用于在p中凸的哈密顿人的背景下，从某种意义上说，$ \ partial $ 2 pp h在每个点都是积极的。我们还将将研究限制在这种情况下。在哈密顿人的扰动之间通过较小的添加潜力和接近一个的积极因素的扰动之间存在密切的关系。实际上，汉密尔顿人H + U和H/（1- u）具有相同的一级能量表面，因此它们在该能量表面上的动力学是彼此的重新对准，这是Maupertuis原理。当H中H中的H（对应于Finsler指标）甚至纤维二次（对应于Riemannian指标）时，此备注尤其重要。在这些情况下，汉密尔顿电势的扰动对应于参数化，以与度量的共形扰动。广泛研究的方面之一是了解与周期轨道相关的返回图在多大程度上可以通过添加较小的潜力来扰动。这种问题在很大程度上取决于他们提出的环境。一些研究最多的环境是在越来越多的难度下，通用媒介领域的扰动，哈密顿系统类内的汉密尔顿系统的扰动，riemannian指标的扰动在里曼尼亚人的范围内，ma {u} {et} { ^} perttrurbations convex hamilton。例如，众所周知，每个向量场都可以扰动到仅具有双曲周期轨道的矢量场，这是库普卡 - 摩尔定理的一部分，请参见[5，13]。在汉密尔顿矢量场的背景下没有这样的结果，但是确实，每个哈密顿量都可以扰动只有非脱位周期性轨道（包括迭代的轨道）的哈密顿量，请参见[11，12]。在黎曼指标的背景下，相同的结果是正确的：每个riemannian指标都可以扰动仅具有非脱位封闭的大地测量学的riemannian指标，这是颠簸的度量定理，请参见[4，2，2，1]。最近，在Ma {n} {é}凸汉密尔顿人的扰动的背景下，研究了这个问题，请参见[9，10]。在[10]中证明了相同的结果：如果h是凸的哈密顿量，而a是h的常规值，则存在任意的电势u，使得能量A处的H + U的所有周期性轨道（包括迭代的轨道）a处于无限量。 [10]中给出的证据实际上与论文中有关Riemannian指标扰动的证据相似。在所有这些证明中，工作非常有用
In the study of Hamiltonian systems on cotangent bundles, it is natural to perturb Hamiltoni-ans by adding potentials (functions depending only on the base point). This led to the definition of Ma{ñ}{é} genericity: a property is generic if, given a Hamiltonian H, the set of potentials u such that H + u satisfies the property is generic. This notion is mostly used in the context of Hamiltonians which are convex in p, in the sense that $\partial$ 2 pp H is positive definite at each points. We will also restrict our study to this situation. There is a close relation between perturbations of Hamiltonians by a small additive potential and perturbations by a positive factor close to one. Indeed, the Hamiltonians H + u and H/(1 -- u) have the same level one energy surface, hence their dynamics on this energy surface are reparametrisation of each other, this is the Maupertuis principle. This remark is particularly relevant when H is homogeneous in the fibers (which corresponds to Finsler metrics) or even fiberwise quadratic (which corresponds to Riemannian metrics). In these cases, perturbations by potentials of the Hamiltonian correspond, up to parametrisation, to conformal perturbations of the metric. One of the widely studied aspects is to understand to what extent the return map associated to a periodic orbit can be perturbed by adding a small potential. This kind of question depend strongly on the context in which they are posed. Some of the most studied contexts are, in increasing order of difficulty, perturbations of general vector fields, perturbations of Hamiltonian systems inside the class of Hamiltonian systems, perturbations of Riemannian metrics inside the class of Riemannian metrics, Ma{ñ}{é} perturbations of convex Hamiltonians. It is for example well-known that each vector field can be perturbed to a vector field with only hyperbolic periodic orbits, this is part of the Kupka-Smale theorem, see [5, 13]. There is no such result in the context of Hamiltonian vector fields, but it remains true that each Hamiltonian can be perturbed to a Hamiltonian with only non-degenerate periodic orbits (including the iterated ones), see [11, 12]. The same result is true in the context of Riemannian metrics: every Riemannian metric can be perturbed to a Riemannian metric with only non-degenerate closed geodesics, this is the bumpy metric theorem, see [4, 2, 1]. The question was investigated only much more recently in the context of Ma{ñ}{é} perturbations of convex Hamiltonians, see [9, 10]. It is proved in [10] that the same result holds : If H is a convex Hamiltonian and a is a regular value of H, then there exist arbitrarily small potentials u such that all periodic orbits (including iterated ones) of H + u at energy a are non-degenerate. The proof given in [10] is actually rather similar to the ones given in papers on the perturbations of Riemannian metrics. In all these proofs, it is very useful to work
