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?"