Geodesic
Features of mappings leading to a central field
Dalʹnevostočnyj matematičeskij žurnal, Tome 20 (2020) no. 1, pp. 58-62.

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For a mechanical system with two degrees of freedom, it is shown that the condition of zeroing the Jacobian map depended on the interaction potential selects a central field that ensures this condition fulfillment. It has been hypothesized that, in the general case, the features of coordinate mappings lead to potentials that admit the existence of an motion integral additional to the energy. 
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M. A. Guzev. Features of mappings leading to a central field. Dalʹnevostočnyj matematičeskij žurnal, Tome 20 (2020) no. 1, pp. 58-62. http://geodesic.mathdoc.fr/item/DVMG_2020_20_1_a5/

[1] V. I. Arnold, Matematicheskie metody klassicheskoi mekhaniki, Editorial URSS, M., 2017