Lagrangian description of fluid motion

[MartinKitty]

Homework Statement

Find velocity, acceleration, stream function and vorticity. Prove that velocity is equal to the acceleration. Functions given:
$$X_1(t,e_1,e_2)= (e^\lambda)^t[e_1cos\omega t+e_2sin(\omega t)]$$
$$X_2(t,e_1,e_2)= (e^-\lambda)^t[-e_1sin\omega t+e_2cos(\omega t)]$$

Homework Equations

$$v(t,e_1,e_2)=\frac{d}{dt} x(t,e_1,e_2)$$
$$a(t,e_1,e_2)=\frac{d}{dt} v(t,e_1,e_2)$$
$$\psi(t,e_1,e_2)=\lambda x_1 x_2$$

The Attempt at a Solution

Calculations of velocity:
$$V_1(t,e_1,e_2)= e_1[(e^\lambda)^t\lambda cos\omega t-(e^\lambda)^t \omega sin\omega t]+e_2[(e^\lambda)^t\lambda sin\omega t+(e^\lambda)^t \omega cos\omega t]$$
$$V_2(t,e_1,e_2)= e_1[(e^-\lambda)^t\lambda sin\omega t-(e^-\lambda)^t \omega cos\omega t]+e_2[(-e^-\lambda)^t\lambda cos\omega t-(e^-\lambda)^t \omega sin\omega t]$$

 

[Asim Riaz]

Calculations of acceleration:A_1(t,e_1,e_2)= e_1[(e^\lambda)^t\lambda^2 cos\omega t-2(e^\lambda)^t \lambda \omega sin\omega t-(e^\lambda)^t \omega^2 cos\omega t]+e_2[(e^\lambda)^t\lambda^2 sin\omega t+2(e^\lambda)^t \lambda \omega cos\omega t+(e^\lambda)^t \omega^2 sin\omega t]A_2(t,e_1,e_2)= e_1[(e^-\lambda)^t\lambda^2 sin\omega t-2(e^-\lambda)^t \lambda \omega cos\omega t-(e^-\lambda)^t \omega^2 sin\omega t]+e_2[(-e^-\lambda)^t\lambda^2 cos\omega t+2(e^-\lambda)^t \lambda \omega sin\omega t+(e^-\lambda)^t \omega^2 cos\omega t]Calculations of stream function:\psi(t,e_1,e_2)=(e^\lambda)^t(e^-\lambda)^t[e_1^2sin(\omega t)+e_2^2cos(\omega t)]Calculations of vorticity:\omega(t,e_1,e_2)=\frac{d}{dt} \psi(t,e_1,e_2)= (e^\lambda)^t(e^-\lambda)^t [e_1^2\omega cos\omega t-e_2^2\omega sin\omega t]To prove that velocity is equal to acceleration, we need to show that V_1(t,e_1,e_2)=A_1(t,e_1,e_2) and V_2(t,e_1,e_2)=A_2(t