I have been given the following Question:
Consider the linear Lagrangian function $L$ in $\mathbb{R}^2$ given by
$$
L(t,x,\dot x)=\alpha(t,x)+\beta(t,x)\dot x,
$$
and the corresponding variational problem with $t_1\leq t\leq t_2$. Write down the Euler-Lagrange equation. (There is more to the question, but those steps are only achievable after finding the EL equation)
Before I finish the rest of the question, i.e. the parts I haven't written down above, I first want to know if I answered the first part correctly.
I assume that $\alpha$ and $\beta$ were constants since they are generally used to denote constants in Hamilton's Principal Function (at least in my study material) as well as the the transformed Hamiltonian.
Thus, my result for the Euler-Lagrange equation: $$ \frac{\partial L}{\partial x}-\frac{d}{dt}\frac{\partial L}{\partial \dot x}=0 $$ is $$ 0-\frac{d}{dt}\beta(t,x)=0\\ \frac{d}{dt}\beta(t,x)=0 $$
was I correct in my assumptions?
Im quite sure that the notation $\alpha(t,x)$ and $\beta(t,x)$ means that these are two functions of $x$ and $t$, not that $\alpha$ and $\beta$ are constant because otherwise the Lagrangian would be a constant times a pair $(x,t)$ indicating it's vector valued when in fact the Lagrangian is always scalar valued.
Thus $\dfrac{\partial L}{\partial x}= \dfrac{\partial \alpha}{\partial x}+\dfrac{\partial \beta}{\partial x}\dot{x}$. The other derivative you calculated is correct.