Geodesic
Force evolutionary billiards and billiard equivalence of the Euler and Lagrange cases
Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 496 (2021), pp. 5-9

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A class of force evolutionary billiards is discovered that realizes important integrable Hamiltonian systems on all regular isoenergy 3-surfaces simultaneously, i.e., on the phase 4-space. It is proved that the well-known Euler and Lagrange integrable systems are billiard equivalent, although the degrees of their integrals are different (two and one).
Keywords: integrable system, billiard, billiard book, Liouville equivalence, Fomenko–Zieschang invariant, evolutionary force billiards, rigid body dynamics. 
@article{DANMA_2021_496_a0,
     author = {V. V. Vedyushkina and A. T. Fomenko},
     title = {Force evolutionary billiards and billiard equivalence of the {Euler} and {Lagrange} cases},
     journal = {Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleni\^a},
     pages = {5--9},
     publisher = {mathdoc},
     volume = {496},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DANMA_2021_496_a0/}
}
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V. V. Vedyushkina; A. T. Fomenko. Force evolutionary billiards and billiard equivalence of the Euler and Lagrange cases. Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 496 (2021), pp. 5-9. http://geodesic.mathdoc.fr/item/DANMA_2021_496_a0/