We define a kinetic and a potential energy such that the principle of stationary action from Lagrangian mechanics yields a Camassa–Holm system (2CH) as the governing equations. After discretizing these energies, we use the same variational principle to derive a semi-discrete system of equations as an approximation of the 2CH system. The discretization is only available in Lagrangian coordinates and requires the inversion of a discrete Sturm–Liouville operator with time-varying coefficients. We show the existence of fundamental solutions for this operator at initial time with appropriate decay. By propagating the fundamental solutions in time, we define an equivalent semi-discrete system for which we prove that there exists unique global solutions. Finally, we show how the solutions of the semi-discrete system can be used to construct a sequence of functions converging to the conservative solution of the 2CH system.

我们定义了一个动能和一个势能，使得拉格朗日力学的静止作用原理产生一个 Camassa-Holm 系统（2CH）作为控制方程。将这些能量离散化后，我们使用相同的变分原理推导出半离散方程组作为 2CH 系统的近似。离散化仅适用于拉格朗日坐标，并且需要对具有时变系数的离散 Sturm-Liouville 算子进行反演。我们展示了该算子在初始时间具有适当衰减的基本解的存在。通过及时传播基本解，我们定义了一个等效的半离散系统，我们证明存在唯一的全局解。最后，