The differential equations of motion of a charged particle in a strong non-uniform magnetic field have the magnetic moment as an adiabatic invariant. This quantity is nearly conserved over long time scales covering arbitrary negative powers of the small parameter, which is inversely proportional to the strength of the magnetic field. The numerical discretisation is studied for a variational integrator that is an analogue for charged-particle dynamics of the Störmer–Verlet method. This numerical integrator is shown to yield near-conservation of a modified magnetic moment and a modified energy over similarly long times. The proofs for both the continuous and the discretised equations use modulated Fourier expansions with state-dependent frequencies and eigenvectors.

中文翻译：

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强磁场中带电粒子动力学变分积分器的长期分析

带电粒子在强非均匀磁场中运动的微分方程具有作为绝热不变量的磁矩。这个量在涵盖小参数的任意负幂的长时间尺度上几乎是守恒的，小参数与磁场强度成反比。针对变分积分器研究数值离散化，该变分积分器类似于 Störmer-Verlet 方法的带电粒子动力学。该数值积分器显示出在类似的长时间内产生近乎守恒的修正磁矩和修正能量。连续方程和离散方程的证明都使用具有状态相关频率和特征向量的调制傅立叶展开。