Understanding the Role of Partial Derivatives in Calculus of Variations

[samgrace]

Hello, here is my problem.[/PLAIN]

http://imgur.com/VAu2sXl

My confusion lies in, why those particular partial derivatives are chosen to be acted upon the auxiliary function and then how they are put together to get the Euler-Lagrange equation?

My guess is that it's related to the turning points of the auxiliary equation, however i don't know why those derivatives are chosen and how they're related to the EL equation.

Sam

 

[Ssnow]

You must interpret $\int_{a}^{b} 12x\cdot y(x)+\left(\frac{\partial}{\partial x} y(x)\right)^{2}dx$ as your action functional $S(t)$ where the time is represented by $t=x, q(t)=y(x), \frac{\partial}{\partial x} y(x)=\dot{q}(t)$ and $f(x,y,y')$ is the lagrangian $\mathcal{L}(t,q,\dot{q})$. In the example derivatives are alculated in order to write the Euler Lagrange equation:

$\frac{\partial}{\partial q}\mathcal{L}-\frac{d}{dt}\frac{\partial}{\partial \dot{q}}\mathcal{L}=\frac{\partial}{\partial y}f-\frac{d}{dx}\frac{\partial}{\partial z}f=0$

They used with abuse of notation $\frac{\partial}{\partial x}=\frac{d}{dx}$ so you solve the Euler Lagrange equation and obtain the extremal function $y(x)$...

remember $z=y'$, I hope in a clarification ...

 

[samgrace]

Oh! Thanks, that's clarified the technique, I can do the rest of worksheet now.