Some results in Lagrangian mechanics
Fiorani, Emanuele
Bollettino dell'Unione Matematica Italiana, Tome 8-A (2005), p. 591-603 / Harvested from 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.

Associamo ad un'equazione dinamica $\xi$ tre differenti connessioni e quindi consideriamo il significato dell'annullarsi della loro curvatura. Alcune peculiarità del caso di equazione dinamica autonoma polinomiale nelle velocità $\dot{q}$ vengono evidenziate. Finalmente, usando le cosiddette condizioni di Helmholtz, indaghiamo un particolare esempio.

Publié le : 2005-10-01
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     author = {Emanuele Fiorani},
     title = {Some results in Lagrangian mechanics},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {8-A},
     year = {2005},
     pages = {591-603},
     zbl = {1150.70011},
     mrnumber = {2182417},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2005_8_8B_3_591_0}
}

Fiorani, Emanuele. Some results in Lagrangian mechanics. Bollettino dell'Unione Matematica Italiana, Tome 8-A (2005) pp. 591-603. http://gdmltest.u-ga.fr/item/BUMI_2005_8_8B_3_591_0/

[1] Abraham, R. - Marsden, J., Foundations of mechanics (II ed.), Benjamin & Cummings, London, 1978. | MR 515141 | Zbl 0393.70001

[2] Giachetta, G. - Mangiarotti, L. - Sardanashvily, G., New Lagrangian and Hamiltonian methods in field theories, World Scientific, Singapore, 1997. | MR 2001723 | Zbl 0913.58001

[3] Saunders, D., The geometry of jet bundles, Cambridge University Press, Cambridge, 1989. | MR 989588 | Zbl 0665.58002

[4] Crampin, M., On the differential geometry of the Euler-Lagrange equations and the inverse problem of Lagrangian dynamics, J. Phys. A: Math. Gen., 14 (1981), 2567-2575. | MR 629315 | Zbl 0475.70022

[5] Crampin, M. - Prince, G. F. - Thompson, G., A geometrical version of the Helmholtz conditions in time-dependent Lagrangian mechanics, J. Phys. A: Math. Gen., 17 (1984), 1437-1447. | MR 748776 | Zbl 0545.58020

[6] Crampin, M. - Martínez, E. - Sarlet, W., Linear connections for systems of second order ordinary differential equations, Ann. Inst. Henri Poincaré, 65 (1996), 223-249. | MR 1411267 | Zbl 0912.58002

[7] Giachetta, G., Jet methods in nonholonomic mechanics, J. Math. Phys., 33 (1992), 1652-1665. | MR 1158984 | Zbl 0758.70010

[8] Havas, P., Four-dimensional formulations of Newtonian mechanics and their relation to the special and general theory of relativity, Rev. Modern Phys., 36 (1964), 938-965. | MR 180329 | Zbl 0123.44101

[9] Massa, E. - Pagani, E., Jet bundle geometry, dynamical connections and the inverse problem of Lagrangian mechanics, Ann. Inst. Henri Poincaré, 61 (1994), 17-62. | MR 1303184 | Zbl 0813.70004