Английская Википедия:Hamiltonian fluid mechanics

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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

<math>\{\rho(\vec{y}),\varphi(\vec{x})\}=\delta^d(\vec{x}-\vec{y})</math>

and the Hamiltonian by:

<math>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>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>

\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>\vec{u}\ \stackrel{\mathrm{def}}{=}\ \nabla \varphi</math> is the velocity and is vorticity-free. The second equation leads to the Euler equations:

<math>\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>\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

Шаблон:Reflist

References

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  1. ↑ Шаблон:Harvnb
  2. ↑ Шаблон:Harvnb
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