Geodesic
Some results in Lagrangian mechanics
Bollettino della Unione matematica italiana, Série 8, 8B (2005) no. 3, pp. 591-603 Cet article a éte moissonné depuis la source Biblioteca Digitale Italiana di Matematica

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We associate to a dynamic equation $\xi$ three different connections and then we consider the meaning of the vanishing of their curvatures. Some peculiarities of the case of autonomous dynamic equation polynomial in the velocities $\dot{q}$ are pointed out. Finally, using the so-called Helmholtz conditions, we investigate a particular example. 
@article{BUMI_2005_8_8B_3_a3,
     author = {Fiorani, Emanuele},
     title = {Some results in {Lagrangian} mechanics},
     journal = {Bollettino della Unione matematica italiana},
     pages = {591--603},
     year = {2005},
     volume = {Ser. 8, 8B},
     number = {3},
     zbl = {1150.70011},
     mrnumber = {MR2182417},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2005_8_8B_3_a3/}
}
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Fiorani, Emanuele. Some results in Lagrangian mechanics. Bollettino della Unione matematica italiana, Série 8, 8B (2005) no. 3, pp. 591-603. http://geodesic.mathdoc.fr/item/BUMI_2005_8_8B_3_a3/