Wheeler-Feynman电动力学的变分原理
Variational principle for the Wheeler-Feynman electrodynamics
摘要：
我们将形式定义的Fokker作用量转化为电磁二体问题的变分原理。我们引入适当定义的边界条件来构造一个有限轨道段的Poincare不变作用泛函。变分原理的边界条件是沿着每个轨迹的端点加上每个端点的光锥内的另一个粒子的相应轨迹段。我们证明了泛函极值的条件是电磁二体问题的隐式状态依赖时滞的混合型中立型方程。我们把泛函放在一个自然的Banach空间，并证明了该泛函是Frechet可微的。我们开发了一种方法来计算一般的C2轨道摄动的第二变分，特别是关于足够大的半径的圆形轨道。我们证明，我们的功能有一个局部极小值在足够大的半径的圆形轨道，在方差与限制开普勒行动，有一个最小值在任意半径的圆形轨道。我们的结果表明，在一定的半径以下的圆轨道成为鞍点极值的分歧。我们给出了一个精确的定义的分布一样的积分的福克行动，并讨论了一个推广到Sobolev空间的轨迹，几乎处处满足的运动方程。最后，我们讨论了以圆弧为边界条件的状态依赖时滞方程解的存在性和存在非平凡螺线管轨道的可能性。
英文摘要：
We adapt the formally-defined Fokker action into a variational principle for the electromagnetic two-body problem. We introduce properly defined boundary conditions to construct a Poincare-invariant-action-functional of a finite orbital segment into the reals. The boundary conditions for the variational principle are an endpoint along each trajectory plus the respective segment of trajectory for the other particle inside the lightcone of each endpoint. We show that the conditions for an extremum of our functional are the mixed-type-neutral-equations with implicit state-dependent-delay of the electromagnetic-two-body problem. We put the functional on a natural Banach space and show that the functional is Frechet-differentiable. We develop a method to calculate the second variation for C2 orbital perturbations in general and in particular about circular orbits of large enough radii. We prove that our functional has a local minimum at circular orbits of large enough radii, at variance with the limiting Kepler action that has a minimum at circular orbits of arbitrary radii. Our results suggest a bifurcation at some radius below which the circular orbits become saddle-point extrema. We give a precise definition for the distributional-like integrals of the Fokker action and discuss a generalization to a Sobolev space of trajectories where the equations of motion are satisfied almost everywhere. Last, we discuss the existence of solutions for the state-dependent delay equations with slightly perturbated arcs of circle as the boundary conditions and the possibility of nontrivial solenoidal orbits.
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