# Functional Derivatives/Euler-Lagrange

• Caramon

## Homework Statement

Hi, I'm working on research and I hit a roadblock with something that should be very simple but I can't solve it because it gets so messy. If anyone can let me know how to do this, it would be greatly appreciated.

I have a functional T:
$$T = \int_{\lambda_{1}}^{\lambda_{2}} sqrt{\sum_{I=1}^{n}} \sum_{i=1}^{d} (\frac{d}{d \lambda}(\sum_{j=1}^{d} s(\lambda) R_{j}^{i}(\lambda)(q_{I}^{j}(\lambda) + a^{j}(\lambda))))^2} d \lambda$$

I need to take functional derivatives with respect to each function defining T and find when they are all concurrently zero. I believe, the Euler-Lagrange equation is able to do this?

I found what $$\frac{\partial{f}}{\partial{g}}$$ is, where g is just a place holder for $$s(\lambda), R_{j}^{i}(\lambda), a^{j}(\lambda), q_{I}^{j}(\lambda)$$

Everytime I tried substituting say, df/dR, or df/ds I would get a huge 22-term, with 5 derivatives, impossible equation that I would have to take f with respected to, where $$T = \int_{\lambda_{1}}^{\lambda_{2}} f d \lambda.$$

I'm looking at this euler-lagrange form:
$$\frac{\partial{f}}{\partial{x}} = \frac{d}{d \lambda} (\left \frac{\partial{f}}{\frac{d}{d \lambda}(\frac{\partial{f}}{\partial{x}})} \right) = 0$$
Where, each member $$s(\lambda), R_{j}^{i}(\lambda), a^{j}(\lambda), q_{I}^{j}(\lambda)$$ are in terms of "x".

Any tips, advice, ideas would be great.