A Deriving Navier-Stokes: Lagrangian & Hamiltonian Methods

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The discussion centers on the possibility of deriving the Navier-Stokes equations using Lagrangian and Hamiltonian methods. Participants express that while there are resources available online, they seek a more structured approach involving generalized coordinates and kinetic energy calculations. It is noted that the Navier-Stokes equations involve energy dissipation, complicating their derivation through these methods. A reference to Peter Constantin's work suggests an Eulerian-Lagrangian approach may be viable. Ultimately, the conversation reflects a desire for clarity on the application of these mathematical frameworks to fluid dynamics.
Hari Seldon
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Is that possible to derive the Navier-Stokes equations with Lagrangian and Hamiltonian methods? If yes, how? and if it is not possible, why?
 
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Did you try Googling this question first? It seems to turn up several hits.
 
Hello, thank you for your reply. Yes, I tried to Google it, but I didn't find what I wanted. I expected an approach like, for example, estabilish the generalized coordinates, calculate the kinetic energy and so on. Finally, that is why I wrote here, I tought that maybe I was thinking in a wrong way.
boneh3ad said:
Did you try Googling this question first? It seems to turn up several hits.
 
Hari Seldon said:
Is that possible to derive the Navier-Stokes equations with Lagrangian and Hamiltonian methods? If yes, how? and if it is not possible, why?
the Navier-Stokes is a system with energy dissipation. The variational principle for the Euler equations is contained in M. Taylor's PDE vol 3
 
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Hari Seldon said:
Is that possible to derive the Navier-Stokes equations with Lagrangian and Hamiltonian methods? If yes, how? and if it is not possible, why?
It seems to some that those equations could be approached with such methods:

An Eulerian-Lagrangian approach to the Navier-Stokes equations. ##-## by Peter Constantin ##-## https://web.math.princeton.edu/~const/xlnsF.pdf
 

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I have recently been really interested in the derivation of Hamiltons Principle. On my research I found that with the term ##m \cdot \frac{d}{dt} (\frac{dr}{dt} \cdot \delta r) = 0## (1) one may derivate ##\delta \int (T - V) dt = 0## (2). The derivation itself I understood quiet good, but what I don't understand is where the equation (1) came from, because in my research it was just given and not derived from anywhere. Does anybody know where (1) comes from or why from it the...
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