论文标题
在缓慢快速的哈密顿系统中的共振中阶段
On phase at a resonance in slow-fast Hamiltonian systems
论文作者
论文摘要
我们考虑一个缓慢快速的哈密顿系统，具有一个快速的角变量（快速相），其频率在缓慢变量（谐振表面）的空间中在某个表面上消失。这种形式的系统出现在高频静电波的影响下的不均匀磁场中带电颗粒的动力学研究中。在快速相系统上平均的轨迹穿过谐振表面。快速阶段使$ \ sim \ frac {1} {\ varepsilon} $转到到达共鸣表面之前（$ \ varepsilon $是问题的小参数）。在到达到达共振时相值的一个渐近公式是在启发式考虑的基础上研究带电粒子动力学的背景下得出的，而没有任何估计其准确性。我们提供了该公式的严格推导，并证明其精度为$ O（\ sqrt \ varepsilon）$（可达对数校正）。数字表明该准确性的估计值是最佳的。
We consider a slow-fast Hamiltonian system with one fast angular variable (a fast phase) whose frequency vanishes on some surface in the space of slow variables (a resonant surface). Systems of such form appear in the study of dynamics of charged particles in inhomogeneous magnetic field under influence of a high-frequency electrostatic waves. Trajectories of the averaged over the fast phase system cross the resonant surface. The fast phase makes $\sim \frac {1}{\varepsilon}$ turns before arrival to the resonant surface ($\varepsilon$ is a small parameter of the problem). An asymptotic formula for the value of the phase at the arrival to the resonance was derived earlier in the context of study of charged particle dynamics on the basis of heuristic considerations without any estimates of its accuracy. We provide a rigorous derivation of this formula and prove that its accuracy is $O(\sqrt \varepsilon)$ (up to a logarithmic correction). Numerics indicate that this estimate for the accuracy is optimal.
