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Noether-type symmetries and conservation laws via partial Lagrangians. (English) Zbl 1121.70014
Summary: We show how one can construct conservation laws of Euler-Lagrange-type equations via Noether-type symmetry operators associated with what we term partial Lagrangians. This is even in the case when a system does not directly have a usual Lagrangian, e.g. scalar evolution equations. These Noether-type symmetry operators do not form a Lie algebra in general. We specify the conditions under which they do form an algebra. Furthermore, we derive conditions under which they are symmetries of Euler-Lagrange-type equations. Examples are given including those that admit a standard Lagrangian such as Maxwellian tail equation, and equations that do not such as the heat and nonlinear heat equations. We also obtain new conservation laws from Noether-type symmetry operators for a class of nonlinear heat equations with more than two independent variables.

MSC:
70H33Symmetries and conservation laws, reverse symmetries, invariant manifolds, etc.
70G65Symmetries, Lie-group and Lie-algebra methods for dynamical systems
References:
[1]Ovsiannikov, L. V., Group Analysis of Differential Equations, Academic Press, New York, 1982.
[2]Ibragimov, N. H., Transformation Groups Applied to Mathematical Physics, Nauka, Moscow, 1983 (English translation by D Reidel, Dordrecht, 1985).
[3]Bluman, G. W. and Kumei, S., Symmetries and Differential Equations, Springer-Verlag, New York, 1989.
[4]Olver, P. J., Application of Lie Groups to Differential Equations, Springer-Verlag, New York, 1993.
[5]Noether, E., ’Invariante variationsprobleme’, Nachr. König. Gesell. Wissen., Göttingen, Math.-Phys. Kl. Heft 2, 1918, 235–257. (English translation in Transport Theory and Statistical Physics 1(3), 1971, 186–207.)
[6]Bessel-Hagen, E., ’Über die Erhaltungssätze der Elektrodynamik’, Mathematische Annalen 84, 1921, 258–276. · Zbl 02603184 · doi:10.1007/BF01459410
[7]Anderson, I. M. and Duchamp, T. E., ’Variational principles for second-order quasi-linear scalar equations’, Journal of Differential Equations 51, 1984, 1–47. · Zbl 0533.49010 · doi:10.1016/0022-0396(84)90100-1
[8]Ibragimov, N. H., CRC Handbook of Lie Group Analysis of Differential Equations Vol 1, Ibragimov, N.H. ed., CRC Press, Boca Raton, Florida, 1994.
[9]Khalique, C. M. and Mahomed, F. M., ’Conservation laws for equations related to soil water equations’, Mathematical Problems in Engineering 26(1), 2005, 141–150. · Zbl 1079.35004 · doi:10.1155/MPE.2005.141
[10]Steudel, H., ’Uber die Zuordnung zwischen Invarianzeigenschaften und Erhal-tungssätzen’, Zeit Naturforsch 17A, 1962, 129–132.
[11]Anco, S. C. and Bluman, G. W., ’Direct construction method for conservation laws of partial differential equations, Part II: General treatment’, European Journal of Applied Mathematics 9, 2002, 567–585.
[12]Wolf, Thomas, ’A comparison of four approaches to the calculation of conservation laws’, European Journal of Applied Mathematics 13, 2002, 129–152. · Zbl 1002.35008 · doi:10.1017/S0956792501004715
[13]Ibragimov, N. H. and Kolsrud, T., ’Lagrangian approach to evolution equations: Symmetries and conservation laws’, Nonlinear Dynamics 36(1), 2004, 29–40. · Zbl 1106.70012 · doi:10.1023/B:NODY.0000034644.82259.1f
[14]Kara, A. H. and Mahomed, F. M., ’Relationship between symmetries and conservation laws’, Int. J. of Theoretical. Phys. 39, 2000, 23–40. · Zbl 0962.35009 · doi:10.1023/A:1003686831523
[15]Kara, A. H., Mahomed, F. M. and Ünal G., ’Approximate symmetries and conservation laws with applications’, Int. J. Theoretical Phys. 38, 1999, 2389–2399. · Zbl 0989.37076 · doi:10.1023/A:1026684004127
[16]Kara, A. H. and Mahomed, F. M., ’A basis of conservation laws for partial differential equations’, J. of Nonlinear Math. Phys. 9, 2002, 60–72. · doi:10.2991/jnmp.2002.9.s2.6
[17]Vinogradov, A. M., ’Local symmetries and conservation laws’, Acta Applied Mathematics 2, 1984, 21–78. · Zbl 0534.58005 · doi:10.1007/BF01405491
[18]Ibragimov, N. H., Kara, A. H. and Mahomed, F. M., ’Lie-Bäcklund and noether symmetries with applications’, Nonlinear Dynamics 15, 1998, 115–136. · Zbl 0912.35011 · doi:10.1023/A:1008240112483
[19]Steinberg, S. and Wolf, K. B., ’Symmetry, conserved quantities and moments in diffusive equations’, Journal of Mathematical Analysis and Applications 80, 1981, 36–45. · Zbl 0471.35034 · doi:10.1016/0022-247X(81)90089-5
[20]Euler, N., Leach, P. G. L., Mahomed, F. M., and Steeb, W.-H., ’Symmetry vector fields and similarity solutions of nonlinear field equation describing the relaxation to a Maxwell distribution’, International Journal of Theoretical Physics 27, 1988, 717–723. · Zbl 0697.35141 · doi:10.1007/BF00669316
[21]Mahomed, F. M., Kara, A. H., and Leach, P. G. L., ’Lie and Noether counting theorems for one-dimensional systems’, Journal of Mathematical Analysis and Applications 178(1), 1993, 116–129. · Zbl 0783.34002 · doi:10.1006/jmaa.1993.1295