free end point variational problem

例句：

1. 1. In the study of fluid dynamics, solving the free end point variational problem is crucial for understanding the behavior of complex flows.
   在流体动力学研究中，解决自由端点变分问题是理解复杂流动行为的关键。
2. 2. The research on the free end point variational problem has led to significant advancements in materials science andengineering.
   对自由端点变分问题的研究表明，在材料科学和工程领域取得了重大进步。
3. 3. The solution to this problem requires a deep understanding of the underlying physics and mathematics, which is why it is often approached with a combination of analytical methods and numerical simulations.
   解决这个问题需要深入理解背后的物理和数学原理，这就是为什么它通常结合使用分析方法和数值模拟的原因。