Help with Lagrangian to Eulerian transformation

Click For Summary
SUMMARY

The discussion focuses on the transformation from Eulerian to Lagrangian motion in continuum mechanics, specifically converting the Eulerian velocity equation V1 = k*z1 into the Lagrangian form z1 = x1*e^(k(t-t0)). The user struggles with solving the differential equation (dz1/dt) + z1^2 = 0, using the initial condition z1 = x1 at t = 0. References to textbooks such as "Introduction to Smooth Manifolds" by Lee are provided for further understanding of the relationship between flows and vector fields.

PREREQUISITES
  • Understanding of Eulerian and Lagrangian frameworks in fluid dynamics
  • Familiarity with differential equations and their solutions
  • Knowledge of initial boundary conditions in mathematical modeling
  • Basic concepts of vector fields and integral curves
NEXT STEPS
  • Study the solution methods for first-order differential equations
  • Learn about the exponential map in the context of differential geometry
  • Explore the relationship between flows and vector fields in continuum mechanics
  • Review the theory of elasticity as discussed in relevant textbooks
USEFUL FOR

Students and professionals in engineering, physics, and applied mathematics who are working with fluid dynamics and continuum mechanics, particularly those interested in the mathematical foundations of motion transformations.

jon8105
Messages
12
Reaction score
0
Ok I have been trying to figure this out for a couple of days now and seem to be stumped. I know it is a fairly simple problem I just can't get it to click! Anyways, here is my problem:

I have a Eulerian velocity of V1 = k*z1 and I want to show that this equals z1 = x1*e^k(t-t0), which is the Lagrangian motion. This is a problem from my continuum mechanics book.

I know that if I solve for the equation (dz1/dt) + z1^2 = 0, with V1 = dz1/dt and initial boundary conditions of z1=x1 at t=0, then I should get the answer, but I am having no luck.

Does anyone know how to convert from Eulerian Velocity, V1 = k*z1, and get the Lagrangian motion, z1 = x1*e^k(t-t0)? Thanks for any help you can provide.
 
Last edited:
Physics news on Phys.org
See any textbook on differential equations (or differential manifolds) which discusses the relationship btween flows, vector fields, integral curves, and the exponential map. For example, Lee, Introduction to Smooth Manifolds. Or see my PF thread, "What is the Theory of Elasticity?"
 

Similar threads

Undergrad Finding vertex of a 3D Triangle on a Plane
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Find 3D point along the line at given distance
  • · Replies 11 ·
Replies
11
Views
7K
Going from Eulerian Velocity to Lagrangian motion
  • · Replies 13 ·
Replies
13
Views
10K
Undergrad Geodesics subject to a restriction
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Mathematical proof for the Lagrangian function
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Discretisation of a PDE in Lagrangian coordinates
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Adding total time derivative to Lagrangian/Canonical Transformations
  • · Replies 2 ·
Replies
2
Views
1K
Graduate The Lagrangian of a free particle ##L=-m \, ds/dt##
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad Types of symmetries in Lagrangian mechanics
  • · Replies 3 ·
Replies
3
Views
2K
Classical mechanics problem for a free particle
  • · Replies 2 ·
Replies
2
Views
2K