Editing PGE/ConservingMomentum (section)

==Time-independent Behavior==
===Lagrangian Frame of Reference (hydrostatic balance)===

If you are riding along with a fluid element &#8212; viewing the system from a ''Lagrangian'' frame of reference &#8212; the velocity {{Math/VAR_VelocityVector01}} of your fluid element will, by definition, remain unchanged over time if, 
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<math>\frac{d\vec{v}}{dt} = 0</math> .
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From the above "Lagrangian Representation" of the Euler equation, this also leads to what is often referred to in discussions of stellar structure as the statement of,
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<span id="HydrostaticBalance"><font color="#770000">'''Hydrostatic Balance'''</font></span><br />

<math>\frac{1}{\rho} \nabla P = - \nabla \Phi</math> .
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That is to say, every fluid element within a star will experience no net acceleration if the gradient of the pressure balances the gradient in the gravitational field throughout the star.

===Eulerian Frame of Reference (steady-state flow field)===

On the other hand, if you are standing at a fixed location in your coordinate frame watching the fluid flow past you &#8212; viewing the system from an ''Eulerian'' frame of reference &#8212; the velocity of the fluid at your location in space will, by definition, always be the same if, 
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<math>\frac{\partial\vec{v}}{\partial t} = 0</math> .
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From the above "Eulerian Representation" of the Euler equation, this condition also implies that a '''steady-state''' velocity field must obey the relation,
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<math>(\vec{v}\cdot \nabla) \vec{v}= - \frac{1}{\rho} \nabla P - \nabla \Phi</math> .
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Or, a '''steady-state''' momentum density field must obey the relation,
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<math>\nabla\cdot [(\rho\vec{v})\vec{v}]= - \nabla P - \rho \nabla \Phi</math>
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