Two-dimensional time-optimal paths of objects moving under the influence of an external force are discussed based on an analysis of Jacobi stability in Finsler space. When the external force on an object can be described by a function of only one variable, the deviation curvature tensor that determines the Jacobi stability of the object’s path can be obtained from the equation of the path. In such cases, the Jacobi stability of the path is represented by the trace of the deviation curvature tensor. The relationship between the Jacobi stability and the type of path is considered for a force that is described by a single-variable trigonometric function. This type of periodic external force induces a path that extends radially and a path along in a specific direction. Then, we consider the time-averaged eigenvalues of the deviation curvature tensor for each type. A large peak in these average values is observed when the type of path changes. Therefore, the Jacobi instability becomes very large at the boundaries between the path types, and the Jacobi stability analysis can be used as the basis of a classification of the path types.

基于芬斯勒空间雅可比稳定性分析，讨论了物体在外力影响下运动的二维时间最优路径。当物体所受的外力只能用一个变量的函数来描述时，就可以从路径方程中得到决定物体路径雅可比稳定性的偏差曲率张量。在这种情况下，路径的雅可比稳定性由偏差曲率张量的迹表示。对于由单变量三角函数描述的力，考虑雅可比稳定性与路径类型之间的关系。这种周期性外力会产生一条径向延伸的路径和一条沿特定方向延伸的路径。然后，我们考虑每种类型的偏差曲率张量的时间平均特征值。当路径类型改变时，可以观察到这些平均值的大峰值。因此，雅可比不稳定性在路径类型之间的边界处变得非常大，雅可比稳定性分析可以作为路径类型分类的基础。