  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.
