In this work, we consider an application of fractional derivatives to realistic physical situations, namely the elastic collision of particles and the nonintegrable diamagnetic Kepler problem. The origin of fractional dynamics can be nonlocal interacting dynamics, memory effects, environments with fractal interacting properties, and relaxation processes, among others. In the case of collisions, considering identical and distinguishable particles, additional solutions appear compared to non-fractional dynamics. For specific velocities of one particle before the collision, several velocities of the other particle are allowed after the collision. Consequently, novel velocity distributions emerge. For the diamagnetic Kepler problem, the fractional dynamic strongly affects the regular and quasi-regular regimes of motion, while the completely chaotic motion remains essentially unaltered. Besides, we derive a fractional momentum-like integral of motion for the pure fractional Kepler problem.

中文翻译：

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分数反磁开普勒问题和弹性碰撞

在这项工作中，我们考虑分数阶导数在现实物理情况中的应用，即粒子的弹性碰撞和不可积的反磁开普勒问题。分数动力学的起源可以是非局域相互作用动力学、记忆效应、具有分形相互作用特性的环境以及松弛过程等。在碰撞的情况下，考虑到相同且可区分的粒子，与非分数动力学相比，会出现额外的解决方案。对于碰撞前一个粒子的特定速度，碰撞后允许另一个粒子的多个速度。因此，出现了新的速度分布。对于反磁开普勒问题，分数动力学强烈影响规则和准规则运动状态，而完全混沌运动基本上保持不变。此外，我们还针对纯分数开普勒问题导出了类似分数动量的运动积分。