AI 中文总结
研究二次阻力与竖直轴马格努斯力下抛体运动的精确转向不变量，通过复数化水平速度并采用路径长度变量，得到航向角固定旋转率、速度指数衰减及曲率半径公式，并解决了无阻力下两球对撞问题。
AI 中文摘要
绕竖直轴旋转的球会受到马格努斯力的侧向推动，因此其地面轨迹会发生弯曲。我们证明，当升力系数和阻力系数为常数时，该问题的水平部分可以用初等函数精确求解。实现这一点的关键步骤是将水平速度表示为复数，并使用路径长度而非时间作为自变量。两种气动力均与速度的平方成正比，因此速度项相互抵消，剩余的是一个不含重力的常系数线性方程。由此得出三个精确结果。航向角每单位路径长度以固定速率旋转，因此无论发射条件如何、无论阻力大小如何，半圈转向所花费的距离始终相同。水平速度随转向角呈指数衰减，衰减速率由阻力与升力之比决定，这使得水平速度矢端图成为对数螺线。地面轨迹的曲率半径等于升力长度乘以飞行路径角的余弦。整个飞行问题还可归结为一个由两个无量纲数控制的标量微分方程。我们利用这一点解决了一个具体问题：两个相同的球以相反的水平方向抛出并带有相反的旋转，在没有阻力的情况下可以迎头相撞，我们给出了闭式解的单参数族；而一旦存在阻力，它们便无法相遇。所有解析结论均与机器精度数值结果进行了核对，代码和数据公开可用。
英文摘要
A ball spinning about a vertical axis is pushed sideways by the Magnus force, so its ground track curves. We show that when the lift and drag coefficients are constant, the horizontal part of this problem is exactly solvable in elementary functions. The step that makes this work is to write the horizontal velocity as a complex number and to use path length rather than time as the independent variable. Both aerodynamic forces are quadratic in speed, so the speed cancels and what remains is a linear equation with constant coefficients in which gravity does not appear. Three exact results follow from it. The heading turns at a fixed rate per unit path length, so a half turn always costs the same distance whatever the launch conditions and whatever the drag. The horizontal speed decays exponentially in the turn angle at a rate fixed by the drag-to-lift ratio, which makes the horizontal hodograph a logarithmic spiral. The radius of curvature of the ground track equals the lift length times the cosine of the flight-path angle. The whole flight problem also collapses to one scalar differential equation controlled by two dimensionless numbers. We use this to settle a concrete question. Two identical balls thrown in opposite horizontal directions with opposite spin can meet head-on when drag is absent, on a one-parameter family we give in closed form, and they fail to meet once drag acts. Every analytic claim is checked against machine-precision numerics, and the code and data are openly available.
Comments19 pages, 4 figures. Code: https://doi.org/10.5281/zenodo.21979028
