Hamiltonian fluid mechanics

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Hamiltonian fluid mechanics is the application of Hamiltonian methods to fluid mechanics. Note that this formalism only applies to nondissipative fluids.

Irrotational barotropic flow[edit]

Take the simple example of a barotropic, inviscid vorticity-free fluid.

Then, the conjugate fields are the mass density field ρ and the velocity potential φ; the Poisson bracket is given by

and the Hamiltonian by:

where e is the internal energy density, as a function of ρ. For this barotropic flow, the internal energy is related to the pressure p by:

where an apostrophe ('), denotes differentiation with respect to ρ.

This Hamiltonian structure gives rise to the following two equations of motion:

where is the velocity and is vorticity-free. The second equation leads to the Euler equations:

after exploiting the fact that the vorticity is zero:

As fluid dynamics is described by non-canonical dynamics, which possess an infinite amount of Casimir invariants, an alternative formulation of Hamiltonian formulation of fluid dynamics can be introduced through the use of Nambu mechanics[1][2]

See also[edit]

Notes[edit]

References[edit]

  • Badin, Gualtiero; Crisciani, Fulvio (2018). Variational Formulation of Fluid and Geophysical Fluid Dynamics - Mechanics, Symmetries and Conservation Laws -. Springer. p. 218. doi:10.1007/978-3-319-59695-2. ISBN 978-3-319-59694-5.
  • Morrison, P.J. (2006). Elsevier (ed.). "Hamiltonian Fluid Mechanics" (PDF). Encyclopedia of Mathematical Physics. 2. Amsterdam. pp. 593–600.
  • Morrison, P. J. (April 1998). "Hamiltonian Description of the Ideal Fluid" (PDF). Reviews of Modern Physics. Austin, Texas. 70 (2): 467–521. Bibcode:1998RvMP...70..467M. doi:10.1103/RevModPhys.70.467.
  • R. Salmon (1988). "Hamiltonian Fluid Mechanics". Annual Review of Fluid Mechanics. 20: 225–256. Bibcode:1988AnRFM..20..225S. doi:10.1146/annurev.fl.20.010188.001301.
  • Shepherd, Theodore G (1990). "Symmetries, Conservation Laws, and Hamiltonian Structure in Geophysical Fluid Dynamics". Advances in Geophysics Volume 32. Advances in Geophysics. 32. pp. 287–338. Bibcode:1990AdGeo..32..287S. doi:10.1016/S0065-2687(08)60429-X. ISBN 9780120188321.
  • Swaters, Gordon E. (2000). Introduction to Hamiltonian Fluid Dynamics and Stability Theory. Boca Raton, Florida: Chapman & Hall/CRC. p. 274. ISBN 1-58488-023-6.
  • Nevir, P.; Blender, R. (1993). "A Nambu representation of incompressible hydrodynamics using helicity and enstrophy". J. Phys. A. 26 (22): 1189–1193. Bibcode:1993JPhA...26L1189N. doi:10.1088/0305-4470/26/22/010.
  • Blender, R.; Badin, G. (2015). "Hydrodynamic Nambu mechanics derived by geometric constraints". J. Phys. A. 48 (10): 105501. arXiv:1510.04832. Bibcode:2015JPhA...48j5501B. doi:10.1088/1751-8113/48/10/105501.