We classify all orthogonal coordinate systems in ${\mathbb{M}}^4$, allowing complete additively separated solutions of the Hamilton–Jacobi equation for a charged test particle in the Liénard–Wiechert field generated by any possible given motion of a point-charge Q. We prove that only the Cavendish–Coulomb field, corresponding to the uniform motion of Q, admits separation of variables, precisely in cylindrical spherical and cylindrical conical-spherical coordinates. We show also that for some fields, the test particle with motion constrained into certain planes admits complete orthogonal separation, and we determine the separable coordinates.

中文翻译：

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Liénard-Wiechert场中带电粒子轨道的Hamilton-Jacobi方程的完全可分性

我们将所有正交坐标系分类 ${\mathbb{中号}}^4$，允许通过点电荷Q的任何给定运动生成的Liénard-Wiechert场中带电测试粒子的Hamilton-Jacobi方程的完全加法分离解。我们证明，只有卡文迪许-库仑场（对应于Q的匀速运动）才允许变量的分离，精确地在圆柱球坐标和圆柱圆锥-球坐标中。我们还表明，对于某些领域，运动受限于某些平面的测试粒子允许完全正交分离，并确定可分离的坐标。