Deriving Navier-Stokes: Lagrangian & Hamiltonian Methods

Click For Summary
SUMMARY

The Navier-Stokes equations can be approached using Lagrangian and Hamiltonian methods, although deriving them directly may present challenges due to energy dissipation characteristics. The variational principle for the Euler equations, as discussed in M. Taylor's "Partial Differential Equations, Volume 3," provides foundational insights into this topic. Additionally, Peter Constantin's work on an Eulerian-Lagrangian approach offers a relevant framework for exploration. This discussion emphasizes the importance of understanding generalized coordinates and kinetic energy calculations in this context.

PREREQUISITES
  • Understanding of Navier-Stokes equations
  • Familiarity with Lagrangian mechanics
  • Knowledge of Hamiltonian dynamics
  • Basic principles of variational calculus
NEXT STEPS
  • Study M. Taylor's "Partial Differential Equations, Volume 3" for insights on variational principles
  • Explore Peter Constantin's Eulerian-Lagrangian approach to the Navier-Stokes equations
  • Learn about energy dissipation in fluid dynamics
  • Investigate generalized coordinates and their applications in Lagrangian mechanics
USEFUL FOR

Mathematicians, physicists, and engineers interested in fluid dynamics, particularly those focused on theoretical approaches to the Navier-Stokes equations.

Hari Seldon
Messages
5
Reaction score
1
Is that possible to derive the Navier-Stokes equations with Lagrangian and Hamiltonian methods? If yes, how? and if it is not possible, why?
 
Physics news on Phys.org
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
 
Reply
  • Like
Likes   Reactions: Hari Seldon, Orodruin and vanhees71
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
 

Similar threads

High School Why doesn't the Navier-Stokes equation have a solution?
  • · Replies 5 ·
Replies
5
Views
1K
Undergrad What year were Navier-Stokes equations introduced?
  • · Replies 11 ·
Replies
11
Views
2K
Undergrad Assumptions for Navier Stokes equations in this system
  • · Replies 3 ·
Replies
3
Views
1K
Graduate Navier-Stokes with spatially varying viscosity
  • · Replies 7 ·
Replies
7
Views
2K
Graduate Hydrostatic equilibrium and Navier-Stokes equations
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Viscosity Term in Navier-Stokes Equation
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad Hamiltonian of the bead rotating on a horizontal stick
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Deriving Navier-Stokes Equation
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Divergence of the Navier-Stokes Equation
  • · Replies 1 ·
Replies
1
Views
5K
Undergrad Outer product of flow velocities in Navier-Stokes equation
  • · Replies 6 ·
Replies
6
Views
2K