How to take partial derivatives of functions whose inputs depend on the same variable?

Question

I am starting to learn about the Calculus of Variations and the Euler-Lagrange equation is extremely confusing to me:

The Euler–Lagrange equation, then, is given by $$L_x(t,q(t),q'(t))-\frac{\mathrm{d}}{\mathrm{d}t}L_v(t,q(t),q'(t)) = 0.$$ where $L_x$ and $L_v$ denote the partial derivatives of $L$ with respect to the second and third arguments, respectively.

Specifically, I don't understand the meaning of: $$L_x = \frac{\partial L(t, q, q')}{\partial q}$$

I'm confused because $q$ and $q'$ are both functions of $t$, but taking a partial derivative requires that we hold other parameters constant.

Yet $t$ is a parameter of $L$ as well. Thus when differentiating $L$ with respect to $q$, we hold $t$ constant.

But if $t$ is constant then doesn't that mean $q$ and $q'$ are constant?
And hence mustn't the derivatives $L_x$ and $L_v$ both be equal to zero?

Answer 1

While we write $L$ as $L(t,q(t),q'(t))$, it is really just a function of three variables. We could just as easily write it as $L(x,y,z)$ and, when evaluating $L$ on a given function $q$, substitute $t$ for $x$, $q(t)$ for $y$, and $q'(t)$ for $z$. However, this could lead to some confusion and lots of extra notation, so it is not usually done.

So, when taking the partial derivative $\frac{\partial L}{\partial q}$, we can just ignore any relations between $q$ and $q'$ and treat them as totally separate variables.

Answer 2

Perhaps better for CS backgrounds: $\partial/\partial t \ F(t,x,y)$ means the derivative of the function with respect to variation in its first slot, the formal parameter $t$. $\mathrm{d}/\mathrm{d} t\ F(t,x,y)$ means the derivative of the function with respect to the system parameter $t$. The difference is between local parameter names (partial differentiation) and global variables (total differentiation).