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The Hamiltonian Field Theory of the Von Mises Wave Equation: Analytical and Computational Issues

Part of: Foundations, constitutive equations, rheology Partial differential equations, initial value and time-dependent initial-boundary value problems Infinite-dimensional Hamiltonian systems

Published online by Cambridge University Press:  16 March 2016

Christian Cherubini  and
Simonetta Filippi
Christian Cherubini*
Affiliation:
Unit of Nonlinear Physics and Mathematical Modeling, Campus Bio-Medico, University of Rome, I-00128, Rome, Italy International Center for Relativistic Astrophysics – I.C.R.A., Campus Bio-Medico, University of Rome, I-00128, Rome, Italy
Simonetta Filippi
Affiliation:
Unit of Nonlinear Physics and Mathematical Modeling, Campus Bio-Medico, University of Rome, I-00128, Rome, Italy International Center for Relativistic Astrophysics – I.C.R.A., Campus Bio-Medico, University of Rome, I-00128, Rome, Italy
*
*Corresponding author. Email addresses:, c.cherubini@unicampus.it (C. Cherubini), s.filippi@unicampus.it (S. Filippi)
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Abstract

The Von Mises quasi-linear second order wave equation, which completely describes an irrotational, compressible and barotropic classical perfect fluid, can be derived from a nontrivial least action principle for the velocity scalar potential only, in contrast to existing analog formulations which are expressed in terms of coupled density and velocity fields. In this article, the classicalHamiltonian field theory specifically associated to such an equation is developed in the polytropic case and numerically verified in a simplified situation. The existence of such a mathematical structure suggests new theoretical schemes possibly useful for performing numerical integrations of fluid dynamical equations. Moreover it justifies possible new functional forms for Lagrangian densities and associated Hamiltonian functions in other theoretical classical physics contexts.

Keywords

47.40.-x02.30.Jr11.10.Ef04.90.+eClassical field theoriesfluid dynamicspartial differential equationsnumerical methods

MSC classification

Secondary: 76A02: Foundations of fluid mechanics 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 37K05: Hamiltonian structures, symmetries, variational principles, conservation laws
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 3 , March 2016 , pp. 758 - 769
Copyright
Copyright © Global-Science Press 2016 

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References

[1]Kambe, T., Elementary Fluid Mechanics, World Scientific, Singapore (2007).Google Scholar
[2]Zakharov, V. E., Musher, S. L. and Rubenchik, A. M., Hamiltonian Approach to the Description of Nonlinear Plasma Phenomena, Phys. Reports, 129, (1985) 285.Google Scholar
[3]Salmon, R., Hamiltonian fluid mechanics, Ann. Rev. Fluid. Mech., 20, (1988) 225.CrossRefGoogle Scholar
[4]Zakharov, V. E. and Kuznetsov, E. A., Hamiltonian formalism for nonlinear waves, Physics Uspekhi, 40, (1997) 1087.Google Scholar
[5]Morrison, P. J., Hamiltonian description of the ideal fluid, Rev. Mod. Phys., 70, (1998) 467.Google Scholar
[6]Seliger, R. L. and Whitham, G. B., Variational Principles in Continuum Mechanics, Proc. Roy. Soc. A, 305, (1968), 1–25.Google Scholar
[7]Bergliaffa, S.E.P, Hibberd, K., Stone, M. and Visser, M., Wave equation for sound in fluids with vorticity, Physica D, 191, (2004) 121.CrossRefGoogle Scholar
[8]Barceló, C., Liberati, S. and Visser, M., Analogue Gravity, Living Rev. Relativity, 8, (2005) 12.CrossRefGoogle ScholarPubMed
[9]Novello, M., Visser, M., and Volovik, G. E. (Editors), Artificial Black Holes, (2002) World Scientific, Singapore.Google Scholar
[10]Unruh, W. and Schutzhold, R., Quantum Analogues: From Phase Transitions to Black Holes and Cosmology, Lect. Notes Phys., 718(2008) Springer, Berlin.Google Scholar
[11]Christensen, R. M., Mechanics of Composite Materials, (2005) Dover.Google Scholar
[12]Kosevich, A.M., Lifshitz, E.M., Landau, L. D. and Pitaevskii, L. P., (Course of Theoretical Physics: Theory of Elasticity (Volume 7), (1986) Butterworth-Heinemann Ed., 3rd edition.Google Scholar
[13]Unruh, W. G., Experimental black hole evaporation, Phys. Rev. Lett., 46, (1981) 1351.Google Scholar
[14]Unruh, W. G., Sonic analogue of black holes and the effects of high frequencies on black hole evaporation, Phys. Rev. D, 51, (1995) 2827.CrossRefGoogle ScholarPubMed
[15]Visser, M., Acoustic black holes: horizons, ergospheres and Hawking radiation, Class. Quantum Grav., 15, (1998) 1767.CrossRefGoogle Scholar
[16]Cherubini, C. and Filippi, S., VonMises’ potential flow wave equation and nonlinear analog gravity, Phys. Rev. D, 84, (2011) 124010.Google Scholar
[17]Cardoso, V., Dias, O.J.C., Lemos, J.P.S. and Yoshida, S., Black-hole bomb and superradiant instabilities, Phys. Rev. D, 70, (2004) 044039.Google Scholar
[18]Berti, E., Cardoso, V. and Lemos, J.P.S., Quasinormal modes and classical wave propagation in analogue black holes, Phys. Rev. D, 70, (2004) 124006.Google Scholar
[19]S., Basak and Majumdar, P., Reflection coefficient for superresonant scattering, Class. Quant. Grav., 20, (2003) 2929.Google Scholar
[20]Basak, S. and Majumdar, P., ‘Superresonance’ from a rotating acoustic black hole, Class. Quant. Grav., 20, (2003) 3907.Google Scholar
[21]Cherubini, C., Federici, F., Succi, S. and Tosi, M. P., Excised acoustic black holes: The scattering problem in the time domain, Phys. Rev. D, 72, (2005) 084016.Google Scholar
[22]Federici, F., Cherubini, C., Succi, S. and Tosi, M. P., Superradiance from hydrodynamic vortices: A numerical study, Phys. Rev. A, 73, (2006) 033604.CrossRefGoogle Scholar
[23]Oliveira, E. S., Dolan, S. R. and Crispino, L.B.C., Absorption of planar waves in a draining bathtub, Phys. Rev. D., 81, (2010) 124013.Google Scholar
[24]Barcelo, C., Liberati, S., Sonego, S. and Visser, M., Hawking-Like Radiation Does Not Require a Trapped Region, Phys. Rev. Lett., 97, (2006) 171301.Google Scholar
[25]Cherubini, C. and Filippi, S., Boundary Conditions for Scattering Problems from Acoustic Black Holes, J.K.P.S., 56, (2010) 1668.Google Scholar
[26]Belgiorno, F., Cacciatori, S. L., Clerici, M., Gorini, V., Ortenzi, G., Rizzi, L., Rubino, E., Sala, V. G. and Faccio, D., Hawking Radiation from Ultrashort Laser Pulse Filaments, Phys Rev. Lett., 105, (2010) 203901.CrossRefGoogle ScholarPubMed
[27]Horstmann, B., Reznik, B., Fagnocchi, S. and Cirac, J. I., Hawking Radiation from an Acoustic Black Hole on an Ion Ring, Phys. Rev. Lett., 104, (2010) 250403.CrossRefGoogle Scholar
[28]Schutzhold, R., Detection Scheme for Acoustic Quantum Radiation in Bose-Einstein Condensates, Phys. Rev. Lett., 97, (2006) 190405.Google Scholar
[29]Fischer, U. R. and Visser, M, Riemannian Geometry of Irrotational Vortex Acoustics, Phys. Rev. Lett., 88, (2002) 110201.Google Scholar
[30]Furuhashi, H., Nambu, Y. and Saida, H., Simulation of an acoustic black hole in a Laval nozzle, Class. Quantum Grav., 23, (2006) 5417.CrossRefGoogle Scholar
[31]Bini, D., Cherubini, C. and Filippi, S., Effective geometries in self-gravitating polytropes, Phys. Rev. D, 78, (2008) 064024.CrossRefGoogle Scholar
[32]Bini, D., Cherubini, C., Filippi, S. and Geralico, A., Effective geometry of the n = 1 uniformly rotating self-gravitating polytrope, Phys. Rev. D, 82, (2010) 044005.Google Scholar
[33]Bini, D., Cherubini, C. and Filippi, S., Effective geometry of a white dwarf, Phys. Rev. D, 83, (2011) 064039.Google Scholar
[34]Cherubini, C. and Filippi, S., Acoustic metric of the compressible draining bathtub, Phys. Rev. D, 84, (2011) 084027.Google Scholar
[35]Cherubini, C. and Filippi, S., An Analog of Einsteins General Relativity Emerging from Classical Finite Elasticity Theory: Analytical and Computational Issues, Commun. Comput. Phys., 14, (2013) 801–818.CrossRefGoogle Scholar
[36]Cherubini, C. and Filippi, S., VonMises potential flow wave equation and nonlinear analog gravity, Phys. Rev. D, 84, 124010 (2011).Google Scholar
[37]Cherubini, C. and Filippi, S., Classical field theory of the Von Mises equation for irrotational polytropic inviscid fluids, J. Phys. A: Math. Theor., 46, 115501 (2013).Google Scholar
[38]Von Mises, R., Mathematical Theory of Compressible Fluid Flow, Dover (2004).Google Scholar
[39]Komzsik, L., Applied Calculus of Variations for Engineers, Second Edition, CRC Press (2008).CrossRefGoogle Scholar
[40]Zwiebach, B., A First Course in String Theory, 2nd Edition, Cambridge University Press (2008).Google Scholar
[41]Corben, H. C. and Stelhe, P., Classical Mechanics 2nd Ed., Dover, (1994).Google Scholar
[42]Goldstein, H., Poole, C. P. Jr. and Safko, J. L., Classical Mechanics, 3rd ed, Addison-Wesley (2001).Google Scholar
[43]Landau, L. D. and Lifshitz, E. M., Mechanics, 3rd ed., Butterworth-Heinemann (1976).Google Scholar
[44]Greiner, W., Reinhardt, J. and Bromley, D. A., Field Quantization, Springer, New York (2008).Google Scholar
[45]Kevorkian, J., Partial Differential Equations: Analytical Solutions and Techniques, 2nd edition, Springer, New York (2000).CrossRefGoogle Scholar
[46]Chandrasekhar, S., An Introduction to the Study of Stellar Structure, Dover Publications (2010).Google Scholar
[47]LeVeque, R. J., Numerical Methods for Conservation Laws, 2nd ed., Birkhauser (2008).Google Scholar
[48]Baumgarte, T. W. and Shapiro, S. L., Numerical Relativity: Solving Einstein's Equations on the Computer, Cambridge University Press (2010).Google Scholar
[49]Zee, A., Quantum Field Theory in a Nutshell 2nd Edition, Princeton University Press (2010).Google Scholar
[50]Lancaster, T. and Blundell, S. J., Quantum field theory for the Gifted Amateur, Oxford University Press (2014).Google Scholar