# Hamiltonian fluid mechanics

**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.