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Keywords:
Lagrange variational problem; Poincaré-Cartan form; symmetry reduction
Lagrange variational problem; Poincaré-Cartan form; symmetry reduction
Summary:
Some open problems appearing in the primary article on the symmetry reduction are solved. A new and quite simple coordinate-free definition of Poincaré-Cartan forms and the substance of divergence symmetries (quasisymmetries) are clarified. The unbeliavable uniqueness and therefore the global existence of Poincaré-Cartan forms without any uncertain multipliers for the Lagrange variational problems are worth extra mentioning.
References:
[1] Bažaňski, S. L.: The Jacobi variational principle revisited. Classical and Quantum Integrability Banach Cent. Publ. 59. Polish Academy of Sciences, Institute of Mathematics, Warsaw (2003), 99-111 J. Grabowski et al. DOI 10.4064/bc59-0-4 | MR 2003718 | Zbl 1082.70008
[2] Chrastinová, V.: The Intransitive Lie Group Actions with Variable Structure Constants. Mathematics, Information Technologies and Applied Sciences 2017 University of Defence, Brno (2017), 141-146 J. Baštinec et al.
[3] Hermann, R.: Differential form methods in the theory of variational systems and Lagrangian field theories. Acta Appl. Math. 12 (1988), 35-78. DOI 10.1007/BF00047568 | MR 0962880 | Zbl 0664.49018
[4] Langerock, B., Cantrijn, F., Vankerschaver, J.: Routhian reduction for quasi-invariant Lagrangians. J. Math. Phys. 51 (2010), Paper No. 022902, 20 pages. DOI 10.1063/1.3277181 | MR 2605045 | Zbl 1309.70019
[5] Olver, P. J., Pohjanpelto, J., Valiquette, F.: On the structure of Lie pseudo-groups. SIGMA, Symmetry Integrability Geom. Methods Appl. 5 (2009), Paper No. 077, 14 pages. DOI 10.3842/SIGMA.2009.077 | MR 2529170 | Zbl 1241.58008
[6] Tryhuk, V., Chrastinová, V.: On the internal approach to differential equations 1. The involutiveness and standard basis. Math. Slovaca 66 (2016), 999-1018. DOI 10.1515/ms-2015-0198 | MR 3567912 | Zbl 06662105
[7] Tryhuk, V., Chrastinová, V.: The symmetry reduction of variational integrals. Math. Bohemica 143 (2018), 291-328. DOI 10.21136/MB.2017.0008-17 | MR 3852296 | Zbl 06940885
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