Two-body relativistic systems in external field
Annales de l'I.H.P. Physique théorique, Volume 50 (1989) no. 2, p. 187-204
@article{AIHPA_1989__50_2_187_0,
     author = {Droz-Vincent, Philippe},
     title = {Two-body relativistic systems in external field},
     journal = {Annales de l'I.H.P. Physique th\'eorique},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {2},
     year = {1989},
     pages = {187-204},
     zbl = {0671.70008},
     mrnumber = {1002819},
     language = {en},
     url = {http://www.numdam.org/item/AIHPA_1989__50_2_187_0}
}
Droz-Vincent, Philippe. Two-body relativistic systems in external field. Annales de l'I.H.P. Physique théorique, Volume 50 (1989) no. 2, pp. 187-204. http://www.numdam.org/item/AIHPA_1989__50_2_187_0/

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We notice that the spring constant is not a first integral, but simply a parameter. According to Lagrange method it is not supposed to vary.

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