Skip to main content
SpringerLink
Log in

An unconstrained Hamiltonian formulation for incompressible fluid flow

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

The equations governing the time evolution of an ideal fluid in material coordinates are expressed as an unconstrained canonical Hamiltonian system. The incompressibility of the flow is consequent upon certain first integrals of the motion. The variable conjugate to the configuration field is not the usual linear momentum, but is instead a quantity that is related to linear momentum through an auxiliary scalar field whose time derivative is the pressure. The definition of the Hamiltonian involves a minimization with respect to this auxiliary field. The method of derivation may be generally applied to obtain unconstrained Hamiltonian descriptions of Lagrangian field equations subject to pointwise constraints.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical aspects of classical and celestial mechanics. In: Arnold, V.I. (ed.) Dynamical Systems III, Encyclopaedia of Mathematical Sciences, vol. 3. Berlin, Heidelberg, New York Springer-Verlag, 1988 (See pp. 31–38)

    Google Scholar 

  2. Benjamin, T.B.: Impulse, flow force and variational principles. IMA J. Appl. Math.32, 3–68 (1984)

    Google Scholar 

  3. Benjamin, T.B., Bridges, T.J.: Reappraisal of the Kelvin-Helmholtz problem. Part 1, Hamiltonian structure. Preprint 1993

  4. Bloch, A., Krishnaprasad, P.S., Marsden, J.E., Ratiu, T.S.: Dissipation induced instabilities. Ann. Inst. H. Poincaré, Anal. Non Linéaire11, 1–54 (1994)

    Google Scholar 

  5. Buttke, T.F.: Velicity methods: Lagrangian numerical methods which preserve the Hamiltonian structure of incompressible fluid flow. In: Beale, J.T., Cottet, G.H., Huberson, S. (eds.) Vortex Flows and Related Numerical Methods. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. vol. 395, pp. 39–57. Dordrecht, Boston, London: Kluwer, 1993

    Google Scholar 

  6. Chorin, A.: Vortex Phase Transitions in 2 1/2 dimensions. Lawrence Berekeley Laboratory Report LBL-34769 (1993)

  7. Dichmann, D.J., Maddocks, J.H., Pego, R.L.: Hamiltonian dynamics of an elastica and the stability of solitary waves. Arch. Rat. Mech. Anal. (submitted)

  8. Dirac, P.A.M.: On generalized Hamiltonian dynamics. Can. J. Math.2, 129–148 (1950)

    Google Scholar 

  9. Holm D.D., Marsden, J.E., Ratiu, T.S.: Hamiltonian structure of continuum mechanics in material, inverse material, spatial and convective representations In: Hamiltonian structure and Lyapunov stability for ideal continuum mechanics. Sem. Math. Sup. vol.100. University of Montreal Press, 1986

  10. Kuz'min, G.A.: Ideal incompressible hydrodynamics in terms of the vortex momentum density. Phys. Lett.96A, 88–90 (1983)

    Google Scholar 

  11. Maddocks, J.H. & Overton, M.L.: Stability theory for dissipatively perturbed Hamiltonian systems. Comm. Pure Appl. Math. (to appear)

  12. Osedelets, V.I.: On a new way of writing the Navier-Stokes equation. The Hamiltonian formalism. Comm. Moscow Math. Soc. (1988) [transl.: Russ. Math. Surveys44, 210–211 (1989)]

  13. Serrin, J.: Mathematical principles of classical fluid mechanics. In: Handbuch der Physik, Vol. VIII/1. Berlin, Heidelberg, New York: Springer-Verlag, 1959, pp. 125–263.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Maryland, 20742, College Park, MD, USA

    John H. Maddocks & Robert L. Pego

  2. Institute for Physical Science and Technology, University of Maryland, 20742, College Park, MD, USA

    John H. Maddocks & Robert L. Pego

Authors
  1. John H. Maddocks
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Robert L. Pego
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by S.-T. Yau

Rights and permissions

Reprints and permissions

About this article

Cite this article

Maddocks, J.H., Pego, R.L. An unconstrained Hamiltonian formulation for incompressible fluid flow. Commun.Math. Phys. 170, 207–217 (1995). https://doi.org/10.1007/BF02099446

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02099446

Keywords

Search

Navigation