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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

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

\( {\displaystyle \{\varphi ({\vec {x}}),\rho ({\vec {y}})\}=\delta ^{d}({\vec {x}}-{\vec {y}})} \)

and the Hamiltonian by:

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

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:

\( e''={\frac {1}{\rho }}p', \)

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

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

\( {\begin{aligned}{\frac {\partial \rho }{\partial t}}&=+{\frac {\partial {\mathcal {H}}}{\partial \varphi }}=-\nabla \cdot (\rho {\vec {u}}),\\{\frac {\partial \varphi }{\partial t}}&=-{\frac {\partial {\mathcal {H}}}{\partial \rho }}=-{\frac {1}{2}}{\vec {u}}\cdot {\vec {u}}-e',\end{aligned}} \)

where \( {\vec {u}}\ {\stackrel {{\mathrm {def}}}{=}}\ \nabla \varphi \) is the velocity and is vorticity-free. The second equation leads to the Euler equations:

\( {\frac {\partial {\vec {u}}}{\partial t}}+({\vec {u}}\cdot \nabla ){\vec {u}}=-e''\nabla \rho =-{\frac {1}{\rho }}\nabla {p} \)

after exploiting the fact that the vorticity is zero:

\( \nabla \times {\vec {u}}={\vec {0}}. \)

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

Luke's variational principle
Hamiltonian field theory


Nevir & Blender 1993

Blender & Badin 2015

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.
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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.

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