We discuss theoretical and numerical aspects of gyrokinetics as a Lagrangian field theory when the field perturbation is introduced into the symplectic part. A consequence is that the field equations and particle equations of motion in general depend on the time derivatives of the field. The most well-known example is when the parallel vector potential is introduced as a perturbation, where a time derivative of the field arises only in the equations of motion, so an explicit equation for the fields may still be written. We will consider the conceptually more problematic case where the time-dependent fields appear in both the field equations and equations of motion, but where the additional term in the field equations is formally small. The conceptual issues were described by Burby (J. Plasma Phys., vol. 82 (3), 2016, 905820304): these terms lead to apparent additional degrees of freedom to the problem, so that the electric field now requires an initial condition, which is not required in low-frequency (Darwin) Vlasov–Maxwell equations. Also, the small terms in the Euler–Lagrange equations are a singular perturbation, and these two issues are interlinked. For well-behaved problems the apparent additional degrees of freedom are spurious, and the physically relevant solution may be directly identified. Because we needed to assume that the system is well behaved for small perturbations when deriving gyrokinetic theory, we must continue to assume that when solving it, and the physical solutions are thus the regular ones. The spurious nature of the singular degrees of freedom may also be seen by changing coordinate systems so the varying field appears only in the Hamiltonian. We then describe how methods appropriate for singular perturbation theory may be used to solve these asymptotic equations numerically. We then describe a proof-of-principle implementation of these methods for an electrostatic strong-flow gyrokinetic system; two basic test cases are presented to illustrate code functionality.

中文翻译：

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求解具有高阶时间依赖性的旋动系统

当将场扰动引入辛部分时，我们将陀螺动力学的理论和数值方面作为拉格朗日场论进行讨论。结果是场方程和运动的粒子方程通常取决于场的时间导数。最著名的例子是当平行向量势作为扰动引入时，场的时间导数只出现在运动方程中，因此仍然可以写出场的显式方程。我们将考虑概念上更成问题的情况，即时间相关场同时出现在场方程和运动方程中，但场方程中的附加项在形式上很小。Burby 描述了概念性问题（J.等离子物理学。, 卷。82 (3), 2016, 905820304)：这些项为问题带来了明显的额外自由度，因此电场现在需要一个初始条件，这在低频 (Darwin) Vlasov-Maxwell 方程中不需要。此外，欧拉-拉格朗日方程中的小项是奇异摄动，这两个问题是相互关联的。对于表现良好的问题，明显的附加自由度是虚假的，并且可以直接识别物理相关的解决方案。因为在推导旋动理论时我们需要假设系统在小扰动下表现良好，所以在求解它时我们必须继续假设，因此物理解是常规解。通过改变坐标系也可以看到奇异自由度的虚假性质，因此变化场仅出现在哈密顿量中。然后我们描述了如何使用适合奇异摄动理论的方法来数值求解这些渐近方程。然后，我们描述了这些方法的原理验证实现，用于静电强流旋动系统；提供了两个基本测试用例来说明代码功能。