Deriving Navier-Stokes: Lagrangian & Hamiltonian Methods

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Discussion Overview

The discussion centers on the possibility of deriving the Navier-Stokes equations using Lagrangian and Hamiltonian methods. Participants explore theoretical approaches and seek clarification on the applicability of these methods to a system characterized by energy dissipation.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions whether it is possible to derive the Navier-Stokes equations using Lagrangian and Hamiltonian methods, seeking both methods and reasoning.
  • Another participant suggests that a search for existing literature on the topic yields several results, implying that the question may have been previously addressed.
  • A participant acknowledges their prior search for information but expresses dissatisfaction with the results, indicating a desire for a more detailed approach involving generalized coordinates and kinetic energy calculations.
  • It is noted that the Navier-Stokes equations involve energy dissipation, and a reference to M. Taylor's work on variational principles for Euler equations is mentioned.
  • A suggestion is made that an Eulerian-Lagrangian approach to the Navier-Stokes equations could be viable, referencing a specific paper by Peter Constantin.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the feasibility of deriving the Navier-Stokes equations using the mentioned methods, and multiple viewpoints regarding the approaches remain present.

Contextual Notes

Participants express uncertainty regarding the applicability of Lagrangian and Hamiltonian methods to a system with energy dissipation, and there are references to specific literature that may or may not address the question satisfactorily.

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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