Abstract
A purely Lagrangian formulation and a direct proof of the separation of variables theorem is given for what is called Stäckel Systems in dynamics and celestial mechanics. The proof is essentially based on some properties of determinants and minors (given in Appendix A). In contrast with the standard literature on the subject, we avoid the use of the Hamiltonian, canonical transformations or the Hamilton-Jacobi equation, by using instead a more elementary approach based on the Lagrangian. In AppendixB we use the Kepler Problem as an illustration of the Lagrangian theory of Stäckel Systems.
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Broucke, R. The Lagrangian theory of Stäckel systems. Celestial Mechanics 25, 185–193 (1981). https://doi.org/10.1007/BF01230519
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DOI: https://doi.org/10.1007/BF01230519