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
- [math]\displaystyle{ \{\rho(\vec{y}),\varphi(\vec{x})\}=\delta^d(\vec{x}-\vec{y}) }[/math]
and the Hamiltonian by:
- [math]\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), }[/math]
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:
- [math]\displaystyle{ e'' = \frac{1}{\rho}p', }[/math]
where an apostrophe ('), denotes differentiation with respect to ρ.
This Hamiltonian structure gives rise to the following two equations of motion:
- [math]\displaystyle{
\begin{align}
\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{align}
}[/math]
where [math]\displaystyle{ \vec{u}\ \stackrel{\mathrm{def}}{=}\ \nabla \varphi }[/math] is the velocity and is vorticity-free. The second equation leads to the Euler equations:
- [math]\displaystyle{ \frac{\partial \vec{u}}{\partial t} + (\vec{u}\cdot\nabla) \vec{u} = -e''\nabla\rho = -\frac{1}{\rho}\nabla{p} }[/math]
after exploiting the fact that the vorticity is zero:
- [math]\displaystyle{ \nabla \times\vec{u}=\vec{0}. }[/math]
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
Notes
References
- 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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