AI 中文总结
研究单自由度哈密顿系统的等正规势，给出解析势函数的等正规性判据，指出与 $x^2/2$ 等正规的势为等时势。
AI 中文摘要
我们考虑具有一个自由度的哈密顿系统：$$\dot x = y, \quad \dot y = -\partial V(x) / \partial x, \qquad x,y\in{\mathbb R}$$，其中势函数 $V: {\mathbb R}\to{\mathbb R}$ 是光滑的。假设原点是椭圆奇点，满足 $V'(0)=0$ 且 $V''(0)>0$。两个势函数 $V_0$ 和 $V_1$ 被称为等正规的，当在原点附近存在一个正则坐标变换，将哈密顿函数 $y^2/2 + V_0$ 转化为 $y^2/2 + V_1$。特别地，任何与 $x^2/2$ 等正规的势函数都是等时的。我们得到了解析势函数的等正规性判据。
英文摘要
We consider the Hamiltonian system with one degree of freedom $$\dot x = y, \quad \dot y = -\partial V(x) / \partial x, \qquad x,y\in{\mathbb R}$$ with the smooth potential $V:{\mathbb R}\to{\mathbb R}$. We assume that the origin is an elliptic singular point: $V'(0)=0$ and $V''(0)>0$. Two potentials $V_0$ and $V_1$ are referred to be isonormal if in a neighborhood of the origin there exists a canonical change of coordinates which transforms the Hamiltonian function $y^2/2 + V_0$ to $y^2/2 + V_1$. In particular, any potential isonormal to $x^2/2$ is isochronous. We obtain a criterium of isonormality for analytic potentials.
Comments9 pages
