Hamiltonian fluid mechanics

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

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

${\displaystyle \{\rho (\overrightarrow{y}),\phi (\overrightarrow{x})\}={\delta }^d(\overrightarrow{x}-\overrightarrow{y})}$

and the Hamiltonian by:

${\displaystyle H=\int {\mathrm{d}}^{d}x{\mathscr{H}}=\int {\mathrm{d}}^{d}x(\frac{1}{2}\rho (\mathrm{\nabla }\phi {)}^2+e(\rho )),}$

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:

${\displaystyle e^{″}=\frac{1}{\rho }{p}^{\prime },}$

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

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

${\displaystyle \begin{array}{rl}\frac{\partial \rho }{\partial t}& =+\frac{\partial {\mathscr{H}}}{\partial \phi }=-\mathrm{\nabla }\cdot (\rho \overrightarrow{u}),\\ \frac{\partial \phi }{\partial t}& =-\frac{\partial {\mathscr{H}}}{\partial \rho }=-\frac{1}{2}\overrightarrow{u}\cdot \overrightarrow{u}-e^{\prime },\end{array}}$

where ${\displaystyle \overrightarrow{u}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\mathrm{\nabla }\phi }$ is the velocity and is vorticity-free. The second equation leads to the Euler equations:

${\displaystyle \frac{\partial \overrightarrow{u}}{\partial t}+(\overrightarrow{u}\cdot \mathrm{\nabla })\overrightarrow{u}=-e^{″}\mathrm{\nabla }\rho =-\frac{1}{\rho }\mathrm{\nabla }p}$

after exploiting the fact that the vorticity is zero:

${\displaystyle \mathrm{\nabla }\times \overrightarrow{u}=\overrightarrow{0}.}$

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]

• Luke's variational principle
• Hamiltonian field theory

• Badin, Gualtiero; Crisciani, Fulvio (2018). Variational Formulation of Fluid and Geophysical Fluid Dynamics - Mechanics, Symmetries and Conservation Laws -. Springer. pp. 218. doi:10.1007/978-3-319-59695-2. ISBN 978-3-319-59694-5. Bibcode: 2018vffg.book.....B. 
• Morrison, P.J. (2006). "Hamiltonian Fluid Mechanics". in Elsevier. Encyclopedia of Mathematical Physics. 2. Amsterdam. pp. 593–600. http://web2.ph.utexas.edu/~morrison/06EMP_morrison.pdf. 
• Morrison, P. J. (April 1998). "Hamiltonian Description of the Ideal Fluid". Reviews of Modern Physics (Austin, Texas) 70 (2): 467–521. doi:10.1103/RevModPhys.70.467. Bibcode: 1998RvMP...70..467M. http://web2.ph.utexas.edu/~morrison/98RMP_morrison.pdf. 
• R. Salmon (1988). "Hamiltonian Fluid Mechanics". Annual Review of Fluid Mechanics 20: 225–256. doi:10.1146/annurev.fl.20.010188.001301. Bibcode: 1988AnRFM..20..225S. https://zenodo.org/record/1063670. 
• Shepherd, Theodore G (1990). "Symmetries, Conservation Laws, and Hamiltonian Structure in Geophysical Fluid Dynamics". Advances in Geophysics Volume 32. 32. pp. 287–338. doi:10.1016/S0065-2687(08)60429-X. ISBN 9780120188321. Bibcode: 1990AdGeo..32..287S. 
• Swaters, Gordon E. (2000). Introduction to Hamiltonian Fluid Dynamics and Stability Theory. Boca Raton, Florida: Chapman & Hall/CRC. pp. 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. doi:10.1088/0305-4470/26/22/010. Bibcode: 1993JPhA...26L1189N. 
• Blender, R.; Badin, G. (2015). "Hydrodynamic Nambu mechanics derived by geometric constraints". J. Phys. A 48 (10): 105501. doi:10.1088/1751-8113/48/10/105501. Bibcode: 2015JPhA...48j5501B. 

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