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[SOLVED] Coupled ODEs from Euler Lagrange eq

dwsmith

Well-known member
Feb 1, 2012
1,673
Given \(F = A(x)u_1^{'2} + B(x)u'_1u'_2 + C(x)u_2^{'2}\).
\[
\frac{\partial F}{\partial u_i} - \frac{d}{dx}\left[\frac{\partial F}{\partial u_i'}\right] = 0
\]
From the E-L equations, I found
\begin{align*}
\frac{d}{dx}\left[2Au_1' + Bu_2'\right] &= 0\\
\frac{d}{dx}\left[2Cu_2' + Bu_1'\right] &= 0\\
2Au_1' + Bu_2' &= D\\
2Cu_2' + Bu_1' &= E
\end{align*}

Just a general question. Is this DE going to be extremely difficult to solve or is it relativily trivial?
 

Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,197
It's going to be relatively trivial. Let $v_{1}=u_{1}'$ and $v_{2}=u_{2}'$. Then it's not actually a DE in the $v_{i}$'s, but just an algebraic equation. This happens because there are no $u_{i}$ terms, but only their derivatives. So you can solve for the $v_{i}$'s using your favorite method, and then at least symbolically integrate the result.