Complex variational calculus with mean of (min, +)-analysis

Michel Gondran, Abdelouahab Kenoufi, Alexandre Gondran


One develops a new mathematical tool, the complex (min, +)-analysis which permits to define a new variational calculus analogous to the classical one (Euler-Lagrange and Hamilton Jacobi equations), but which is well-suited for functions defined from C^n to C. We apply this complex variational calculus to Born-Infeld theory of electromagnetism and show why it does not exhibit nonlinear effects.


Variational Calculus, Lagrangian, Hamiltonian, Action, Euler-Lagrange and Hamilton-Jacobi equations, complex (min, +)-analysis , Maxwell’s equations, Born-Infeld theory.

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[BB96] I. Bialynicki-Birula. Photon wave function. E. Wolf (eds.), Progress in Optics, Elsevier, Amsterdam, 80:1588–1590, 1996.

[BI33] M. Born and L. Infeld. Foundations of the new field theory. Nature,

:1004, 1933.

[Bor37] M. Born. Théorie non linéaire du champ électromagnétique. Ann. Inst. H. Poincaré, 7(1):155–261, 1937.

[Bre98] D. Breche. Bps states of the non-abelian born-infeld action. Phys. Lett., 442(B):117–124, 1998.

[BW99] M. Born and E. Wolf. Principles of optics. Cambridge University, 7th

edition, 1999.

[dM44] P.L. de Maupertuis. Accord de différentes lois de la nature qui avaient jusqu’ici paru incompatibles. Mémoires de l’Académie Royale des Sciences, Paris, pages 417–426, 1744.

[dM46] P.L. de Maupertuis. Les lois du mouvement et du repos déduites d’un principe métaphysique. Mémoire Académie Berlin, page 267, 1746.

[Ein99] A. Einstein. A generalized theory of gravitation. Rev. Mod. Phys., 20:35–39, 1999.

[Eul44] L. Euler. Methodus Inveniendi Lineas Curvas Maximi Minive Proprietate Gaudentes. Bousquet, Lausanne et Geneva, 1744.

[Eva98] L. C. Evans. Partial differential equations. Graduate Studies in Mathematics 19, American Mathematical Society, pages 123–124, 1998.

[Gon96] M. Gondran. Analyse minplus. C. R. Acad. Sci. Paris, 323:371–375, 1996.

[Gon99] M. Gondran. Convergences de fonctions à valeurs dans R k et analyse minplus complexe. C. R. Acad. Sci. Paris, 329:783–788, 1999.

[Gon01] M. Gondran. Analyse minplus complexe. C. R. Acad. Sci. Paris, 333:592–598, 2001.

[HG02] J. Safko H. Goldstein, C. Poole. Classical mechanics. 3rd Edition, San Francisco : Addison-Wesley, 2002.

[Jac99] J.D. Jackson. Classical Electrodynamics. John Wiley and Sons, 3rd edition, 1999.

[JRF73] W. Greiner J. Rafelski and L.P. Fulcher. Superheavy elements and non-linear electrodynamics. Nuovo Cimento, 13(B):135, 1973.

[Lag88] J.L. Lagrange. Mécanique Analytique. Gauthier-Villars, Paris, translated by V. Vagliente and A. Boissonade (Klumer Academic, Dordrecht, 2001), 2nd edition, 1888.

[L.L70] E.Lifchitz L.Landau. Classical theory of fields. Mir, Moscow, 1970.

[LL76] L.D. Landau and E.M. Lifshitz. Mechanics, Course of Theoretical Physics. Buttreworth-Heinemann, London, 1976.

[Mas87] V. Maslov. Operational calculus. Mir, Moscow, 1987.

[MB34] L. Infeld M. Born. Foundations of the new field theory. Proc. Roy. Soc. London, 144(A):425–451, 1934.

[MG03] R. Hoblos M. Gondran. Complex calculus of variations. Kybernetika Max-Plus special issue, 39(2):677–680, 2003.

[MG08] M. Minoux M. Gondran. Graphs, Dioids and Semirings. Springer, 2008.

[MG14] A. Kenoufi M. Gondran. Numerical calculations of Hölder exponents for the Weierstrass functions with (min,+)-wavelets. Trends in Applied and Computational Mathematics, 15(3), 2014.

[MG16] T. Lehner M. Gondran, A. Kenoufi. Multi-fractal analysis for riemann

serie and mandelbrot binomial measure with (min, +)-wavelets. Trends in

Applied and Computational Mathematics, 17(2):247–263, 2016.

[Min08] H. Minkowski. Die grundgleichungen für die elektromagnetischen vorgänge in bewegten körpern. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 53:111, 1908.

[P.A58] P.A.M.Dirac. The principles of Quantum Theory. Clarendon Press, Oxford, 4th edition, 1958.

[Sch81] L.S. Schulmann. Techniques and Applications of Path Integration. John Wiley & Sons, New York, 1981.

[SGS98] F.A. Schaposnik S. Gonorazky, C. Nunez and G. Silva. Bogomol’nyi bounds and the supersymmetric born-infeld theory. Nucl. Phys., B 531(B):168–184, 1998.

[Sil07] L. Silberstein. Nachtrag zur abhandlung über electromagnetische grundgleichungen in bivektorieller behandlung. Ann. Phys. Lpz., 24:783, 1907.

[Sil24] L. Silberstein. The theory of relativity. Macmillan and Co Ltd, London, 1924.

[Tho98] L. Thorlaciu. Born-infeld string as a boundary conformal field theory. Phys. Rev. Lett., 80:1588–1590, 1998.

[VM92] S.N. Samborski V. Maslov. Idempotent analysis. Advances in Soviet Math-ematics, American Mathematical Society, 13, 1992.


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