Practicing with solved problems is the fastest way to understand Lagrangian mechanics. Below are some of the best, freely available PDF resources online.
If you are looking for specific types of problems (e.g., rigid body dynamics or electromagnetic Lagrangian), please let me know, and I can narrow down the resources!
Problem 2: Mass on a Frictionless Inclined Wedge (Atwood-style Variation) A wedge of mass and incline angle sits on a frictionless horizontal floor. A block of mass
A classic problem demonstrating coupled oscillations and more complex Euler-Lagrange application.