论文标题
循环动作的局部最小值$ 1/r^α$电势和三体问题的图形解决方案
Local minimality properties of circular motions in $1/r^α$ potentials and of the figure-eight solution of the 3-body problem
论文作者
论文摘要
我们首先考虑到周期性边界条件的变异问题，并简要回忆到某些足够的条件，以定期解决Euler-Lagrange方程的定期解决方案是方向性，弱或强大的局部最小化器。然后，我们将理论应用于开普勒问题的圆形轨道，其电位$ 1/r^α，\，α> 0 $。通过使用数值计算，我们表明圆形解决方案是$α> 1 $的强大局部最小化器，而它们是$α\ in（0,1）$的鞍点。此外，我们表明，对于（1,2）$ $α\，在周期性曲线上，该动作的全球最小化器在$ 2 $的周期性曲线上相对于原点可以实现，可以在非碰撞和非圆形解决方案上实现。之后，我们考虑了三体问题的图八解，我们表明它是一组特定的对称周期循环的强大局部最小化器。
We first take into account variational problems with periodic boundary conditions, and briefly recall some sufficient conditions for a periodic solution of the Euler-Lagrange equation to be either a directional, a weak, or a strong local minimizer. We then apply the theory to circular orbits of the Kepler problem with potentials of type $1/r^α, \, α> 0$. By using numerical computations, we show that circular solutions are strong local minimizers for $α> 1$, while they are saddle points for $α\in (0,1)$. Moreover, we show that for $α\in (1,2)$ the global minimizer of the action over periodic curves with degree $2$ with respect to the origin could be achieved on non-collision and non-circular solutions. After, we take into account the figure-eight solution of the 3-body problem, and we show that it is a strong local minimizer over a particular set of symmetric periodic loops.
