本文介绍了高阶导数的动力学系统的基本概念；推导了含高阶导数的拉格朗日函数所满足的高阶拉格朗日方程；从高阶导数的动力学系统的拉格朗日函数导出了对应的哈密顿量，并以 Pais-Uhlenbeck 振子为例展示了哈密顿量的 Ostrogradsky 不稳定性 (即高阶导数的动力学系统的哈密顿量没有下界); 研究了二阶朗之万方程，并从二阶朗之万方程出发推导出了对应的福克普朗克方程；证明了玻尔兹曼分布 P ∝ e^−H/kBT 满足二阶导数的福克普朗克方程以及更高阶的福克普朗克方程。

This paper introduces the basic concepts of dynamical systems with higher-order derivatives. The higher-order Lagrangian equation satisfied by the higher-order Lagrangian function with higher-order derivatives is derived. The corresponding Hamiltonian is derived from the Lagrange function of the dynamical system of the higher derivative, and the Ostrogradsky instability of the Hamiltonian is shown by the Pais-Uhlenbeck oscillator. The second-order Langevin equation was studied, and the corresponding Fokker Planck equation was derived from the second-order Langevin equation. It is proved that the Boltzmann distribution P ∝ e^−H/kBT satisfies the Fokker Planck equation of the second derivative and the higher derivative Fokker Planck equation。