The Stäckel theorem in the Lagrangian formalism and the use of local times

G. F. Torres del Castillo


We show that the conditions for the separability of the Hamilton-Jacobi equation given by the Stäckel theorem imply that, making use of the elementary Lagrangian formalism, one can find $n$ functionally independent constants of motion, where $n$ is the number of degrees of freedom. We also show that this result can be linked to the fact that the Lagrangian for a system of this class is related to the sum of $n$ one-dimensional Lagrangians, if one makes use of multiple local times.


Stäckel's theorem; Lagrangian formulation; fictitious time.

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L.A. Pars, A Treatise on Analytical Dynamics (Wiley, New York, 1965; reprinted by Ox Bow Press, 1979), Sec. 18.2.

D.T. Greenwood, Classical Dynamics (Prentice-Hall, Englewood Cliffs, New Jersey, 1977; reprinted by Dover, 1997), Sec. 5-3.

H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, MA, 1980), Sec. 10-4 and Appendix D.

A.M. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras, Vol. I (Birkhauser, Basel, 1990), Sect. 2.3.

G.F. Torres del Castillo, The use of fictitious time in Lagrangian mechanics, Rev. Mex. Fís. E (to appear).

D.T. Greenwood, op. cit., p. 208.

G.F. Torres del Castillo, An Introduction to Hamiltonian Mechanics (Springer, Cham, 2018), Example 6.5. 1007/978-3-319-95225-3

L.A. Pars, op. cit., Sec. 18.3.

L.A. Pars, op. cit., Sec. 27.2.

H. Goldstein, op. cit., Sec. 8-6.



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