In this challenging analytic survey, we present a new approach for solving equations of the dynamics of a charged particle (describing its motion in variable, time-dependent electromagnetic field) which has been applied earlier in various fields of mechanics for solving equations of hydrodynamics, Euler–Poisson equations of rigid body rotation and even in celestial mechanics (solution of equations of small body’s motion in the CR3BP problem near librations points): a new type of the solving procedure is implemented in all of these equations as well as in case of solving momentum equation for the aforementioned dynamics of a charged particle, determined by Lorentz force in non-relativistic case. Meanwhile, in each case the system of momentum equations has been successfully solved analytically. The main result of the current research should be formulated as follows: the analytic algorithm is pointed out for solving momentum equation, which has been reduced to the analytical solution of three nonlinear ODEs with respect to the components of velocity of the particle. Moreover, absolutely new partial analytical solutions have been obtained for the special cases of magnetic field (with zero electric field’s components). In addition to this, we conclude that system of Lorentz momentum equations for dynamics of a charged particle has not the analytical presentation of solution in case of nonzero, time-dependent electric field.

在这项具有挑战性的分析性调查中，我们提出了一种求解带电粒子动力学方程的新方法（描述其在可变的，随时间变化的电磁场中的运动），该方法已早先应用于各种力学领域，以求解流体力学方程，刚体旋转甚至在天体力学中的Euler-Poisson方程（在CR3BP问题中解放点附近的小物体运动方程的解）：在所有这些方程中以及在在非相对论情况下，由洛伦兹力确定带电粒子动力学的动量方程式。同时，在每种情况下，动量方程组都已成功解析。当前研究的主要结果应表述如下：提出了一种求解动量方程的解析算法，该算法已针对粒子速度分量简化为三种非线性ODE的解析解。而且，绝对新对于磁场（具有零电场分量）的特殊情况，已经获得了部分解析解。除此之外，我们得出结论，在非零时变电场的情况下，带电粒子动力学的Lorentz动量方程组没有解析解的解析表示。