A real-world application of Lagrangian mechanics: The circular restricted 3-Body Problem

The application of advanced classical mechanics (specifically, the Lagrangian and/or Hamiltonian approaches) to real-world problems is not always straightforward, despite the mathematical rigor and elegance of this field. The objective of this work is to connect the relatively abstract concept of Lagrangian mechanics into the more familiar realm of orbital mechanics. We present a brief, yet complete and self-contained, derivation of the Jacobi integrals using the Lagrangian formulation. The result provides a useful example of how advanced classical mechanics may prove simpler and more efficient than standard methods using Newton's second law. We complement these results with two case studies in our Solar system: the Circular Restricted 3-Body Problems around Pluto and Charon and around Earth and the Moon. These examples provide useful illustrations of both unstable and stable Lagrange points. We further propose a representation of zero-velocity curves, which separate the permissible and forbidden regions of motion for the object, given a constant of the motion. This representation allows an interpretation of Jacobi constants using energy density. The use of S.I. units over normalization in our case studies departs from the traditional examples to offer a more pedagogical and intuitive approach to this problem.