论文标题
动作角度坐标以在具有恒定梯度的直磁场中运动
Action-angle coordinates for motion in a straight magnetic field with constant gradient
论文作者
论文摘要
直磁场中带电粒子的运动$ {\ bf b} = b（y）\，\ wh {\ sf z} $，具有恒定垂直梯度的固定，以椭圆形函数和积分术语来求解。该运动可以根据$ y $轴的定期运动以及沿$ x $轴的漂移运动进行分解。周期性运动可以描述为被困在$ y $中的对称四分之一电位的粒子。从规范坐标$（y，p_ {y}）$转换为动作角坐标$（j，θ）$，根据生成函数$ s（θ，j）$明确求解，该函数用jacobi Elliptic函数表示。弱常数电场$ {\ bf e} = e_ {0} \，\ wh {\ sf y} $的存在会引起四分势的不对称组件，并且相关的周期性运动被逆向驱动求解到二阶。
The motion of a charged particle in a straight magnetic field ${\bf B} = B(y)\,\wh{\sf z}$ with a constant perpendicular gradient is solved exactly in terms of elliptic functions and integrals. The motion can be decomposed in terms of a periodic motion along the $y$-axis and a drift motion along the $x$-axis. The periodic motion can be described as a particle trapped in a symmetric quartic potential in $y$. The canonical transformation from the canonical coordinates $(y,P_{y})$ to the action-angle coordinates $(J,θ)$ is solved explicitly in terms of a generating function $S(θ,J)$ that is expressed in terms of Jacobi elliptic functions. The presence of a weak constant electric field ${\bf E} = E_{0}\,\wh{\sf y}$ introduces an asymmetric component to the quartic potential, and the associated periodic motion is solved perturbatively up to second order.
