Connecting Euler and Lagrange systems as nonlocally related systems of dynamical nonlinear elasticity

• Autorzy: Bluman G. , Ganghoffer J. F.
• Języki publikacji: EN

Abstrakty

EN

Nonlocally related systems for the Euler and Lagrange systems of twodimensional dynamical nonlinear elasticity are constructed. Using the continuity equation, i.e., conservation of mass of the Euler system to represent the density and Eulerian velocity components as the curl of a potential vector, one obtains the Euler potential system that is nonlocally related to the Euler system. It is shown that the Euler potential system also serves as a potential system of the Lagrange system. As a consequence, a direct connection is established between the Euler and Lagrange systems within a tree of nonlocally related systems. This extends the known situation for one-dimensional dynamical nonlinear elasticity to two spatial dimensions.

Słowa kluczowe

EN

dynamical nonlinear elasticity; potential variables; nonlocally related systems; symmetries; conservation laws; gauge constraints

• Wydawca: Instytut Podstawowych Problemów Techniki PAN
• Czasopismo: Archives of Mechanics
• Rocznik: 2011
• Tom: Vol. 63, nr 4
• Strony: 363--382
• Opis fizyczny: Bibliogr. 19 poz.

Twórcy

autor

Bluman G.

autor

Ganghoffer J. F.

• Mathematics Department Room 121, Mathematics Building 1984 Mathematics Road Vancouver, BC Canada V6T 1Z2,

Bibliografia

• 1. S. Anco, G. Bluman, Nonlocal symmetries and nonlocal conservation laws of Maxwell’s equations, J. Math. Phys., 38, 3508–3532, 1997.
• 2. S. Anco, G. Bluman, Direct construction method for conservation laws of partial differentia equations. Part I: Examples of conservation law classifications, European J. Appl. Math., 13, 545–566, 2002.
• 3. S. Anco, G. Bluman, Direct construction method for conservation laws of partial differentia equations. Part II: General treatment, European J. Appl. Math., 13, 567–585, 2002.
• 4. S. Anco, D. The, Symmetries, conservation laws, and cohomology of Maxwell’s equations using potentials, Acta. Appl. Math., 89, 1–52, 2005.
• 5. G. Bluman, A.F. Cheviakov, S. Anco, Applications of Symmetry Methods to Partial Differential Equations, Springer, New York 2010.
• 6. G. Bluman, A.F. Cheviakov, J.F. Ganghoffer, Nonlocally related PDE systems for one-dimensional nonlinear elastodynamics, J. Eng. Math., 62, 203–221, 2008.
• 7. G. Bluman, A.F. Cheviakov, J.F. Ganghoffer, On the nonlocal symmetries, group invariant solutions and conservation laws of nonlinear dynamical compressible elasticity, Proc. IUTAM Symposium on ‘Progress in the Theory and Numerics of Configurational Mechanics’, 20-25/10/2009. Springer, 2009, 14 pp.
• 8. O.I. Bogoyavlenskij, Infinite symmetries of the ideal MHD equilibrium equations, Phys. Lett. A, 291, 256–264, 2001.
• 9. A. Cheviakov, G. Bluman, Multidimensional partial differential equations systems: Generating new systems via conservation laws, potentials, gauges, subsystems, J. Math. Phys., 51, 103521, 2010.
• 10. A. Cheviakov, G. Bluman, Multidimensional partial differential equations systems: Nonlocal symmetries, nonlocal conservation laws, exact solutions, J. Math. Phys., 51, 103522, 2010.
• 11. A.F. Cheviakov, Computation of fluxes of conservation laws, J. Eng. Math., DOI: 10.1007/s10665-009-9307-x (2009).
• 12. A.F. Cheviakov, GeM software package for computation of symmetries and conservation laws of differential equations, Comp. Phys. Comm., 176, 48–61, 2007.
• 13. F. Galas, Generalized symmetries for the ideal MHD equations, Phys. D, 63, 87–98,1993.
• 14. C.O. Horgan, J.H. Murphy, Lie group analysis and plane strain bending of cylindrical sectors for compressible nonlinearly elastic materials, IMA J. Appl. Math., 70, 80–91,2005.
• 15. R. Ogden, Non-linear Elastic Deformations, Ellis Harwood, 1997.
• 16. R.O. Popovych, N.M. Ivanova, Hierarchy of conservation laws of diffusion-convection equations, J. Math. Phys., 46, 043502, 2005.
• 17. H.D. Wahlquist, F.B. Estabrook, Prolongation structures of nonlinear evolution equations, J. Math. Phys., 16, 1–7, 1975.
• 18. V.V. Zharinov, Conservation laws of evolution systems [in Russian], Teor. Mat. Fiz., 68, 163–171, 1986; English transl.: Theor. Math. Phys., 68, 745–751, 1987.
• 19. V.V. Zharinov, Geometrical Aspects of Partial Differential Equations, World Sci., Singapore, 1992,.