We show that non-relativistic scaling of the collisionless Vlasov–Maxwell system implies the existence of a formal invariant slow manifold in the infinite-dimensional Vlasov–Maxwell phase space. Vlasov–Maxwell dynamics restricted to the slow manifold recovers the Vlasov–Poisson and Vlasov–Darwin models as low-order approximations, and provides higher-order corrections to the Vlasov–Darwin model more generally. The slow manifold may be interpreted to all orders in perturbation theory as a collection of formal Vlasov–Maxwell solutions that do not excite light waves, and are therefore ‘dark’. We provide a heuristic lower bound for the time interval over which Vlasov–Maxwell solutions initialized optimally near the slow manifold remain dark. We also show how the dynamics on the slow manifold naturally inherits a Hamiltonian structure from the underlying system. After expressing this structure in a simple form, we use it to identify a manifestly Hamiltonian correction to the Vlasov–Darwin model. The derivation of higher-order terms is reduced to computing the corrections of the system Hamiltonian restricted to the slow manifold.

中文翻译：

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Vlasov-Maxwell 到暗慢流形的哈密顿量减少

我们表明，无碰撞 Vlasov-Maxwell 系统的非相对论标度意味着在无限维 Vlasov-Maxwell 相空间中存在形式不变的慢流形。受限于慢流形的 Vlasov-Maxwell 动力学将 Vlasov-Poisson 和 Vlasov-Darwin 模型恢复为低阶近似，并更一般地为 Vlasov-Darwin 模型提供高阶校正。慢流形可以解释为微扰理论中的所有阶，作为不激发光波的正式 Vlasov-Maxwell 解的集合，因此是“黑暗的”。我们为 Vlasov-Maxwell 解在慢速流形附近初始化为最佳的时间间隔提供了一个启发式下限，该时间间隔保持黑暗。我们还展示了慢流形上的动力学如何自然地从底层系统继承哈密顿结构。在以简单的形式表达这个结构之后，我们用它来识别对 Vlasov-Darwin 模型的明显哈密顿校正。高阶项的推导简化为计算系统哈密顿量限制在慢流形上的修正。