Hamiltonian fluid mechanics

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

## Irrotational barotropic flow

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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

{ ρ ( y -> ) , φ ( x -> ) } = δ d ( x -> − y -> ) {\displaystyle \\{\rho ({\vec {y}}),\varphi ({\vec {x}})\\}=\delta ^{d}({\vec {x}}-{\vec {y}})}

and the Hamiltonian by:

H = ∫ d d x H = ∫ d d x ( 1 2 ρ ( ∇ φ ) 2 + e ( ρ ) ) , {\displaystyle H=\int \mathrm {d} ^{d}x{\mathcal {H}}=\int \mathrm {d} ^{d}x\left({\frac {1}{2}}\rho (\nabla \varphi )^{2}+e(\rho )\right),}

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:

e ″ = 1 ρ p ′ , {\displaystyle e''={\frac {1}{\rho }}p',}

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

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

∂ ρ ∂ t = + ∂ H ∂ φ = − ∇ ⋅ ( ρ u -> ) , ∂ φ ∂ t = − ∂ H ∂ ρ = − 1 2 u -> ⋅ u -> − e ′ , {\displaystyle {\begin{aligned}{\frac {\partial \rho }{\partial t}}&=+{\frac {\partial {\mathcal {H}}}{\partial \varphi }}=-\nabla \cdot (\rho {\vec {u}}),\\\\{\frac {\partial \varphi }{\partial t}}&=-{\frac {\partial {\mathcal {H}}}{\partial \rho }}=-{\frac {1}{2}}{\vec {u}}\cdot {\vec {u}}-e',\end{aligned}}}

where  u -> = d e f ∇ φ {\displaystyle {\vec {u}}\ {\stackrel {\mathrm {def} }{=}}\ \nabla \varphi } is the velocity and is vorticity-free. The second equation leads to the Euler equations:

∂ u -> ∂ t + ( u -> ⋅ ∇ ) u -> = − e ″ ∇ ρ = − 1 ρ ∇ p {\displaystyle {\frac {\partial {\vec {u}}}{\partial t}}+({\vec {u}}\cdot \nabla ){\vec {u}}=-e''\nabla \rho =-{\frac {1}{\rho }}\nabla {p}}

after exploiting the fact that the vorticity is zero:

∇ × u -> = 0 -> . {\displaystyle \nabla \times {\vec {u}}={\vec {0}}.}

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

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* Luke's variational principle
* Hamiltonian field theory

## Notes

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1. ^ Nevir & Blender 1993
2. ^ Blender & Badin 2015

## References

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* Badin, Gualtiero; Crisciani, Fulvio (2018). Variational Formulation of Fluid and Geophysical Fluid Dynamics - Mechanics, Symmetries and Conservation Laws -. Springer. p. 218\. Bibcode:2018vffg.book.....B. doi:10.1007/978-3-319-59695-2. ISBN 978-3-319-59694-5. S2CID 125902566.
* Morrison, P.J. (2006). "Hamiltonian Fluid Mechanics" (PDF). In Elsevier (ed.). Encyclopedia of Mathematical Physics. Vol. 2\. Amsterdam. pp. 593–600.`{{cite encyclopedia}}`: CS1 maint: location missing publisher (link)
* Morrison, P. J. (April 1998). "Hamiltonian Description of the Ideal Fluid" (PDF). Reviews of Modern Physics. 70 (2). Austin, Texas: 467–521\. Bibcode:1998RvMP...70..467M. doi:10.1103/RevModPhys.70.467. hdl:2152/61087.
* 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. Vol. 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. S2CID 119661148.