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A modified formal Lagrangian formulation for general differential equations

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Abstract

In this paper, we propose a modified formal Lagrangian formulation by introducing dummy dependent variables and prove the existence of such a formulation for any system of differential equations. The corresponding Euler–Lagrange equations, consisting of the original system and its adjoint system about the dummy variables, reduce to the original system via a simple substitution for the dummy variables. The formulation is applied to study conservation laws of differential equations through Noether’s Theorem and in particular, a nontrivial conservation law of the Fornberg–Whitham equation is obtained by using its Lie point symmetries. Finally, a correspondence between conservation laws of the incompressible Euler equations and variational symmetries of the relevant modified formal Lagrangian is shown.

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References

  1. Anco, S.C.: On the incompleteness of Ibragimov’s conservation law theorem and its equivalence to a standard formula using symmetries and adjoint-symmetries. Symmetry 9, 33 (2017)

    Article  MathSciNet  Google Scholar 

  2. Anco, S.C., Bluman, G.W.: Direct construction method for conservation laws of partial differential equations. Part I: examples of conservation law classifications. Eur. J. Appl. Math. 13, 545–566 (2002)

    Article  Google Scholar 

  3. Anco, S.C., Bluman, G.W.: Direct construction method for conservation laws of partial differential equations. Part II: general treatment. Eur. J. Appl. Math. 13, 567–585 (2002)

    Article  Google Scholar 

  4. Anderson, I.M.: The Variational Bicomplex. Utah State University, Utah (1989)

    Google Scholar 

  5. Atherton, R.W., Homsy, G.M.: On the existence and formulation of variational principles for nonlinear differential equations. Stud. Appl. Math. 54, 31–60 (1975)

    Article  MathSciNet  Google Scholar 

  6. Bocharov, A.V., Chetverikov, V.N., Duzhin, S.V., Khor’kova, N.G., Krasil’shchik, I.S., Samokhin, A.V., Torkhov, Yu.N., Verbovetsky, A.M., Vinogradov, A.M.: Symmetries and Conservation Laws for Differential Equations of Mathematical Physics. AMS Publications, Providence, RI (1999)

    Book  Google Scholar 

  7. Bridges, T.J.: Multi-symplectic structures and wave propagation. Math. Proc. Camb. Phil. Soc. 121, 147–190 (1997)

    Article  MathSciNet  Google Scholar 

  8. Clarkson, P.A., Mansfield, E.L., Priestley, T.J.: Symmetries of a class of nonlinear third-order partial differential equations. Math. Comput. Model. 25, 195–212 (1997)

    Article  MathSciNet  Google Scholar 

  9. Cotter, C.J., Holm, D.D., Hydon, P.E.: Multisymplectic formulation of fluid dynamics using the inverse map. Proc. Roy. Soc. A 463, 2671–2687 (2007)

    Article  MathSciNet  Google Scholar 

  10. Fornberg, B., Whitham, G.B.: A numerical and theoretical study of certain nonlinear wave phenomena. Phil. Trans. Roy. Soc. Lond. A 289, 373–404 (1978)

    Article  MathSciNet  Google Scholar 

  11. Gandarias, M.L.: Weak self-adjoint differential equations. J. Phys. A Math. Theor. 44, 262001 (2011)

    Article  Google Scholar 

  12. Göktaş, Ü., Hereman, W.: Symbolic computation of conserved densities for systems of nonlinear evolution equations. J. Symb. Comput. 24, 591–622 (1997)

    Article  MathSciNet  Google Scholar 

  13. Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integrators: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  14. Hashemi, M.S., Haji-Badali, A., Vafadar, P.: Group invariant solutions and conservation laws of the Fornberg-Whitham equation. Z. Naturforsch. A 69, 489–496 (2014)

    Article  Google Scholar 

  15. Hereman, W.: Symbolic computation of conservation laws of nonlinear partial differential equations in multi-dimensions. Int. J. Quantum Chem. 106, 278–299 (2006)

    Article  Google Scholar 

  16. Hydon, P.E.: Symmetry Methods for Differential Equations: A Beginner’s Guide. Cambridge University Press, Cambridge (2000)

    Book  Google Scholar 

  17. Ibragimov, N.H.: A new conservation theorem. J. Math. Anal. Appl. 333, 311–328 (2007)

    Article  MathSciNet  Google Scholar 

  18. Ibragimov, N.H.: Nonlinear self-adjointness and conservation laws. J. Phys. A Math. Theor. 44, 432002 (2011)

    Article  Google Scholar 

  19. Ibragimov, N.H., Khamitova, R.S., Valenti, A.: Self-adjointness of a generalized Camassa-Holm equation. Appl. Math. Comput. 218, 2579–2583 (2011)

    MathSciNet  MATH  Google Scholar 

  20. Kara, A.H., Mahomed, F.M.: Noether-type symmetries and conservation laws via partial Lagrangians. Nonlinear Dyn. 45, 367–383 (2006)

    Article  MathSciNet  Google Scholar 

  21. Kosmann-Schwarzbach, Y.: The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century. Springer-Verlag, New York (2011)

    Book  Google Scholar 

  22. Kraus, M.: Variational integrators for inertial magnetohydrodynamics. Phys. Plasmas 25, 082307 (2018)

    Article  Google Scholar 

  23. Kraus, M., Maj, O.: Variational integrators for nonvariational partial differential equations. Physica D 310, 37–71 (2015)

    Article  MathSciNet  Google Scholar 

  24. Kraus, M., Tassi, E., Grasso, D.: Variational integrators for reduced magnetohydrodynamics. J. Comput. Phys. 321, 435–458 (2016)

    Article  MathSciNet  Google Scholar 

  25. Kupershmidt, B.A.: Geometry of jet bundles and the structure of Lagrangian and Hamiltonian formalisms. Lect. Notes Math. 775, 162–218 (1980)

    Article  MathSciNet  Google Scholar 

  26. Mansfield, E.L.: A Practical Guide to the Invariant Calculus. Cambridge University Press, Cambridge (2010)

    Book  Google Scholar 

  27. Mansfield, E.L., Rojo-Echeburúa, A., Hydon, P.E., Peng, L.: Moving frames and Noether’s finite difference conservation laws I. Trans. Math. Appl. 3, tnz004 (2019)

    MathSciNet  Google Scholar 

  28. Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry, 2nd edn. Springer-Verlag, New York (1999)

    Book  Google Scholar 

  29. Marsden, J.E., West, M.: Discrete mechanics and variational integrators. Acta Numer. 10, 357–514 (2001)

    Article  MathSciNet  Google Scholar 

  30. Marsden, J.E., Patrick, G.W., Shkoller, S.: Multisymplectic geometry, variational integrators, and nonlinear PDEs. Commun. Math. Phys. 199, 351–395 (1998)

    Article  MathSciNet  Google Scholar 

  31. Noether, E.: Invariante Variationsprobleme, Nachr. König. Gesell. Wissen. Göttingen Math. Phys. Kl. 2, 235–257 (1918). English transl.: Transport Theory Stat. Phys. 1, 186–207 (1971)

  32. Obata, K.: Formal Lagrangians Applied to Partial Differential Equations. Undergraduate Thesis, Keio University (2020)

  33. Olver, P.J.: Applications of Lie Groups to Differential Equations, 2nd edn. Springer-Verlag, New York (1993)

    Book  Google Scholar 

  34. Olver, P.J.: Equivalence, Invariants, and Symmetry. Cambridge University Press, Cambridge (1995)

    Book  Google Scholar 

  35. Peng, L.: From Differential to Difference: The Variational Bicomplex and Invariant Noether’s Theorems. Ph.D. Thesis, University of Surrey (2013)

  36. Peng, L.: Self-adjointness and conservation laws of difference equations. Commun. Nonlinear Sci. Numer. Simul. 23, 209–219 (2015)

    Article  MathSciNet  Google Scholar 

  37. Peng, L.: Symmetries, conservation laws, and Noether’s theorem for differential-difference equations. Stud. Appl. Math. 139, 457–502 (2017)

    Article  MathSciNet  Google Scholar 

  38. Saunders, D.J.: The Geometry of Jet Bundles. LMS Lecture Note Series 142, Cambridge University Press, Cambridge (1989)

  39. Vinogradov, A.M.: The ${\mathscr {C}}$-spectral sequence, Lagrangian formalism, and conservation laws. I. The linear theory. J. Math. Anal. Appl. 100, 1–40 (1984)

    Article  MathSciNet  Google Scholar 

  40. Vinogradov, A.M.: The ${\mathscr {C}}$-spectral sequence, Lagrangian formalism, and conservation laws. II. The nonlinear theory. J. Math. Anal. Appl. 100, 41–129 (1984)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was partially supported by JSPS KAKENHI Grant Number JP20K14365, JST-CREST Grant Number JPMJCR1914, Keio Gijuku Academic Development Funds, and Keio Gijuku Fukuzawa Memorial Fund. L. Peng is adjunct faculty member at the School of Mathematics and Statistics, Beijing Institute of Technology, China, and adjunct researcher at the Waseda Institute for Advanced Study, Waseda University, Japan. We thank the anonymous reviewers for their valuable comments.

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  1. Department of Mechanical Engineering, Keio University, Yokohama, 223-8522, Japan

    Linyu Peng

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Peng, L. A modified formal Lagrangian formulation for general differential equations. Japan J. Indust. Appl. Math. 39, 573–598 (2022). https://doi.org/10.1007/s13160-022-00500-7

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