Help with Lagrangian to Eulerian transformation

[jon8105]

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.

 

[Chris Hillman]

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?"

 

[tacman]

The Lagrangian to Eulerian transformation is a common problem in continuum mechanics and it can be solved using the chain rule and the inverse transformation. Let's break down the steps to solve this problem:

Step 1: Define the Eulerian and Lagrangian coordinates

Eulerian coordinates are used to describe the motion of a fluid or a solid in a fixed reference frame. In this case, z1 represents the position of a particle in the fluid or solid at a given time t.

Lagrangian coordinates, on the other hand, are used to describe the motion of a particle in a moving reference frame. In this case, x1 represents the initial position of the particle at time t0.

Step 2: Determine the relationship between the Eulerian and Lagrangian coordinates

The relationship between the Eulerian and Lagrangian coordinates can be expressed as:

z1 = x1 + ξ(t)

Where ξ(t) is the displacement of the particle from its initial position at time t0.

Step 3: Apply the chain rule

Using the chain rule, we can express the Eulerian velocity V1 as:

V1 = dz1/dt = dx1/dt + dξ/dt

Step 4: Substitute the given Eulerian velocity

Substituting the given Eulerian velocity V1 = k*z1 into the equation from step 3, we get:

k*z1 = dx1/dt + dξ/dt

Step 5: Solve for dξ/dt

Solving for dξ/dt, we get:

dξ/dt = k*z1 - dx1/dt

Step 6: Substitute the initial boundary conditions

Substituting the initial boundary conditions of z1 = x1 at t=0, we get:

dξ/dt = k*x1 - dx1/dt

Step 7: Integrate both sides

Integrating both sides with respect to time, we get:

ξ(t) = k*x1*t - x1 + C

Where C is the constant of integration.

Step 8: Substitute back into the relationship between Eulerian and Lagrangian coordinates

Substituting the value of ξ(t) into the relationship between Eulerian and Lagrangian coordinates, we get:

z1 = x1 + k*x1*t - x1 + C

Simplifying, we get:

z1