Using a variational method, we exhibit a surprisingly simple periodic orbit for the newtonian problem of three equal masses in the plane. The orbit has zero angular momentum and a very rich symmetry pattern. Its most surprising feature is that the three bodies chase each other around a fixed eight-shaped curve. Setting aside collinear motions, the only other known motion along a fixed curve in the inertial plane is the ``Lagrange relative equilibrium" in which the three bodies form a rigid equilateral triangle which rotates at constant angular velocity within its circumscribing circle. Our orbit visits in turns every ``Euler configuration" in which one of the bodies sits at the midpoint of the segment defined by the other two (Figure 1). Numerical computations by Carles Simó, to be published elsewhere, indicate that the orbit is ``stable" (i.e. completely elliptic with torsion). Moreover, they show that the moment of inertia I(t) with respect to the center of mass and the potential U(t) as functions of time are almost constant.
Non-existence of periodic solutions in fractional-order dynamical systems and a remarkable difference between integer and fractional-order derivatives of periodic functions
Periodic orbits for three particles with finite angular momentum
本文提出了一种新的数值方法来计算动力系统的周期轨道的迭代过程，这是直接基于经典力学的作用量积分。在$1/r^{p+1}$形式的两两吸引力作用下，得到了三个等质量粒子在具有有限总角动量的共同周期轨道上的平面运动的新解.结果表明，对于$-2 < p \le 0$，拉格朗日1772圆解是一个复杂的对称轨道的极限情况。这个轨道的演变和另一个最近发现的一个形状的数字8的角动量的范围内进行了调查。扩展到n个相同质量的粒子和三个不同质量的粒子也进行了简要讨论。
