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Integrating factors, adjoint equations and Lagrangians. (English) Zbl 1102.34002

A new concept of an adjoint equation is used for construction of a Lagrangian for any system of differential equations. The method presented is illustrated by considering several equations traditionally regarded as equations without Lagrangians. Noether’s theorem is applied to the Maxwell equations.

MSC:

34A05 Explicit solutions, first integrals of ordinary differential equations
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
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[1] Bessel-Hagen, E., Über die Erhaltungssätze der Elektrodynamik, Math. Ann., 84, 258-276 (1921) · JFM 48.0877.02
[2] Courant, R.; Hilbert, D., Methods of Mathematical Physics. Vol. II: Partial Differential Equations, by R. Courant (1989), Interscience, Wiley: Interscience, Wiley New York · Zbl 0729.35001
[3] Ibragimov, N. H., Lie-Bäcklund groups and conservation laws, Dokl. Akad. Nauk SSSR. Dokl. Akad. Nauk SSSR, Soviet Math. Dokl., 17, 5, 1242-1246 (1976), English transl. · Zbl 0358.35052
[4] Ibragimov, N. H., Sur l’équivalence des équations d’évolution, qui admettent une algébre de Lie-Bäcklund infinie, C. R. Acad. Sci. Paris Sér. I, 293, 657-660 (1981) · Zbl 0482.35047
[5] Ibragimov, N. H., Transformation Groups Applied to Mathematical Physics (1985), Nauka: Nauka Moscow: Riedel: Nauka: Nauka Moscow: Riedel Dordrecht, English transl.
[6] Ibragimov, N. H., Elementary Lie Group Analysis and Ordinary Differential Equations (1999), Wiley: Wiley Chichester · Zbl 1047.34001
[7] Ibragimov, N. H.; Kolsrud, T., Lagrangian approach to evolution equations: Symmetries and conservation laws, Nonlinear Dynamics, 36, 1, 29-40 (2004) · Zbl 1106.70012
[8] Landau, L. D.; Lifshitz, E. M., Field Theory. Course of Theoretical Physics, vol. 2, The Classical Theory of Fields (1971), Fizmatgiz: Pergamon: Fizmatgiz: Pergamon New York, English transl.
[9] Noether, E., Invariante Variationsprobleme, Königliche Gesellschaft der Wissenschaften, Göttingen Math. Phys. Kl.. Königliche Gesellschaft der Wissenschaften, Göttingen Math. Phys. Kl., Transport Theory and Statistical Physics, 1, 3, 186-207 (1971), English transl.
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