Sorry, I won't give any equations because I have not dabbled enough with the maths of passive components in AC. I can somewhat understand the first order and a lot less in second derivative equations.
But what we have to understand is what is causal and not. While this might seem like an un-resolvable philosophical question, this one is not. The equation and description is not causal. It is a relationship. Which means as long as the voltage a given value at a given time, then the current should exist and we have this certain value.
You can turn the question into a causal one. You can ask, "OK, I have a reliable voltage source (which can supply any amount of stream of charges to maintain the voltage across it) that is AC, what's the current?". Which is one side of the question. One can also ask, "If I have a current source that is AC, what would the voltage across the capacitor be?".
By the way, a current source charging a capacitor is hardly ever given consideration, but if we were to implement a setup like that. You can have a high enough voltage source (to provide the peak current) then have a transistor (BJT or MOSFET) provide the current. Incidentally, you can also say "the voltage drop across the transistor vary to create the voltage across it and the rest across the capacitor", as the equation tells us. You can see it from different points of view.
Equations only give us relationship and because we can plug-in numbers, value, but hardly ever the context.
I'm gonna get a lot of fire for posting this simple idea, but if it wasn't for this, I wouldn't have resolved a system that I was solving in a purely mathematical sense a while back.