Jacobi integral: A real-world application of the Lagrangian formulation

The applicability of advanced classical mechanics (viz., the Lagrangian and/or Hamiltonian approaches) to real-world problems may not always seem straightforward, despite the mathematical rigor and elegance of this field. Here, we present a proof of the Jacobi integral using the Lagrangian formulation as a viable alternative to the usual demonstration using Newton's second law. The result represents a useful example of how advanced classical mechanics can provide a significant advantage over standard methods (i.e., Newton's laws). We conclude with an illustration of the Jacobi integral in our Solar system: the Circular Restricted 3-Body Problem (CR3BP) around Pluto and Charon.