论文标题
通用1D机械系统的动作角度变量
Action-angle Variables for Generic 1D Mechanical Systems
论文作者
论文摘要
我们考虑一个1D机械系统$$ \ bar {\ Mathtt H}（\ Mathtt p，\ Mathtt Q）= \ Mathtt P^2+\ bar {\ Mathtt G}（\ Mathtt G}（\ Mathtt Q）in Action-Mathle and in Action-angle variable in Action-Angtt p，\ nathtt p，\ mathtt p，\ mathtt p，\ mathtt q） $2π$ - 周期性分析功能，具有非退化临界点。然后，我们考虑$ \ bar {\ mathtt h} $的$ \ bar {\ mathtt H}^*的少量分析扰动*（\ mathtt p，\ mathtt p，\ mathtt q; \ hat {\ hat {\ mathtt p}） （\ MATHTT P，\ MATHTT Q; \ HAT {\ MATHTT P}）=：\ MATHTT P^2 + {\ MATHTT G}^*（\ MATHTT P，\ MATHTT P，\ MATHTT Q; \ HAT Q; \ HAT {\ MATHTT P} P} P}） g}^*$可能取决于$ \ mathtt p $的操作，也取决于参数$ \ hat {\ mathtt p} $（“ the Adiabatic Actions”）;的确，这是有限的维度机械系统的形式，在平均快速角度平均并忽略了指数级的剩余部分之后，接近精确的简单共振，请参见[5]。最多可以将$ {\ Mathtt H}^*$的相位空间分为有限数量的分隔数和椭圆形/双曲点。在每个连接的组件上，我们都会执行集成系统的（Arnold-Liouville）符号动作角度转换。我们对这种集成转换的分析性属性进行了完整而定量的描述，特别是估计了这种转换与$ \ bar {\ mathtt H} $的集成转换的不同之处；比较下面的定理6.1。
We consider a 1D mechanical system $$\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 perturbed potential $ {\mathtt G}^*$ may depend on the action $\mathtt P$ and also on parameters $\hat{\mathtt P}$ ("the adiabatic actions"); indeed, this is the form of a finite dimensional mechanical system close to an exact simple resonance after averaging over fast angles and disregarding the exponentially small remainder, see [5]. Up to a finite number of separatrices and elliptic/hyperbolic points the phase space of ${\mathtt H}^*$ is divided into a finite number of open connected components foliated by invariant circles. On every connected component we perform a (Arnold-Liouville) symplectic action-angle transformation which integrates the system. We give a complete and quantitative description of the analyticity properties of such integrating transformations, estimating, in particular, how such transformations differ from the integrating transformation for $\bar {\mathtt H}$; compare Theorem 6.1 below.
