Geodesic
The inverse problem in the calculus of variations: new developments
Communications in Mathematics, Tome 29 (2021) no. 1, pp. 131-149

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We deal with the problem of determining the existence and uniqueness of Lagrangians for systems of $n$ second order ordinary differential equations. A number of recent theorems are presented, using exterior differential systems theory (EDS). In particular, we indicate how to generalise Jesse Douglas's famous solution for $n=2$. We then examine a new class of solutions in arbitrary dimension $n$ and give some non-trivial examples in dimension 3.
We deal with the problem of determining the existence and uniqueness of Lagrangians for systems of $n$ second order ordinary differential equations. A number of recent theorems are presented, using exterior differential systems theory (EDS). In particular, we indicate how to generalise Jesse Douglas's famous solution for $n=2$. We then examine a new class of solutions in arbitrary dimension $n$ and give some non-trivial examples in dimension 3.
Classification : 37J06, 49N45, 58A15, 70H03
Keywords: Inverse problem in the calculus of variations; Helmholtz conditions; Exterior differential systems; Lagrangian system. 
Do, Thoan; Prince, Geoff. The inverse problem in the calculus of variations: new developments. Communications in Mathematics, Tome 29 (2021) no. 1, pp. 131-149. http://geodesic.mathdoc.fr/item/COMIM_2021_29_1_a7/
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[1] Aldridge, J.E.: Aspects of the Inverse Problem in the Calculus of Variations. 2003, La Trobe University, Australia,

[2] Aldridge, J.E., Prince, G. E., Sarlet, W., Thompson, G.: An EDS approach to the inverse problem in the calculus of variations. J. Math. Phys., 47, 2006, | DOI | MR

[3] Anderson, I., Thompson, G.: The inverse problem of the calculus of variations for ordinary differential equations. Memoirs Amer. Math. Soc., 98, 473, 1992, | DOI | MR | Zbl

[4] Bryant, R.L., Chern, S.S., Gardner, R.B., Goldschmidt, H.L., Griffiths, P.A.: Exterior Differential Systems. 1991, Springer, | MR

[5] Crampin, M., Martínez, E., Sarlet, W.: Linear connections for systems of second--order ordinary differential equations. Ann. Inst. H. Poincaré Phys. Théor., 65, 1996, 223-249, | MR

[6] Crampin, M., Prince, G.E., Sarlet, W., Thompson, G.: The inverse problem of the calculus of variations: separable systems. Acta Appl. Math., 57, 1999, 239-254, | DOI | MR

[7] Crampin, M., Prince, G.E., Thompson, G.: A geometric version of the Helmholtz conditions in time-dependent Lagrangian dynamics. J. Phys. A: Math. Gen., 17, 1984, 1437-1447, | DOI | MR

[8] Crampin, M., Sarlet, W., Martínez, E., Byrnes, G.B., Prince, G.E.: Toward a geometrical understanding of Douglas's solution of the inverse problem in the calculus of variations. Inverse Problems, 10, 1994, 245-260, | DOI | MR

[9] Do., T.: The Inverse Problem in the Calculus of Variations via Exterior Differential Systems. 2016, La Trobe University, Australia, | MR

[10] Do, T., Prince, G.E.: New progress in the inverse problem in the calculus of variations. Diff. Geom. Appl., 45, 2016, 148-179, | DOI | MR

[11] Douglas, J.: Solution of the inverse problem of the calculus of variations. Trans. Am. Math. Soc., 50, 1941, 71-128, | DOI | MR | Zbl

[12] Helmholtz, H.: Über der physikalische Bedeutung des Princips der kleinsten Wirkung. J. Reine Angew. Math., 100, 1887, 137-166, | MR

[13] Henneaux, M.: On the inverse problem of the calculus of variations. J. Phys. A: Math. Gen., 15, 1982, L93-L96, | DOI | MR

[14] Henneaux, M., Shepley, L. C.: Lagrangians for spherically symmetric potentials. J. Math. Phys., 23, 1988, 2101-2107, | DOI | MR

[15] Hirsch, A.: Die Existenzbedingungen des verallgemeinterten kinetischen Potentialen. Math. Ann., 50, 1898, 429-441, | DOI | MR

[16] Jerie, M., Prince, G.E.: Jacobi fields, linear connections for arbitrary second order ODE's. J. Geom. Phys., 43, 2002, 351-370, | DOI | MR

[17] Krupková, O., Prince, G.E.: Second order ordinary differential equation in jet bundles, the inverse problem of the calculus of variation. 2008, in: HandBook of Global Analysis, Elsevier, | MR

[18] Massa, E., Pagani, E.: Jet bundle geometry, dynamical connections,, the inverse problem of Lagrangian mechanics. Ann. Inst. Henri Poincaré, Phys. Theor., 61, 1994, 17-62, | MR

[19] Morandi, G., Ferrario, C., Vecchio, G. Lo, Marmo, G., Rubano, C.: The inverse problem in the calculus of variations, the geometry of the tangent bundle. Phys. Rep., 188, 1990, 147-284, | DOI | MR

[20] Sarlet, W.: The Helmholtz conditions revisited. A new approach to the inverse problem of Lagrangian dynamics. J. Phys. A: Math. Gen., 15, 1982, 1503-1517, | DOI | MR | Zbl

[21] Sarlet, W., Crampin, M., Martínez, E.: The integrability conditions in the inverse problem of the calculus of variations for second-order ordinary differential equations. Acta Appl. Math., 54, 1998, 233-273, | DOI | MR

[22] Sarlet, W., Thompson, G., Prince, G.E.: The inverse problem of the calculus of variations: the use of geometrical calculus in Douglas's analysis. Trans. Amer. Math. Soc., 354, 2002, 2897-2919, | DOI | MR

[23] Sonin, N. Ya.: On the definition of maximal, minimal properties. Warsaw Univ. Izvestiya, 1--2, 1886, 1-68,