If you don't have a middle and an end, you don't have a beginning. I don't mean that you have to work out all the details of the middle and the end before you begin, because you can work those out as you go if that is how you work. What I mean is that the beginning sets up the middle and the end so if your don't have a middle and an end in mind, you don't know what the beginning is supposed to set up.
At the heart of every story there is a choice about values. The protagonist will be brought to a point where they have to make a fundamental moral choice. In order to grasp something new, they will have to open their hand and let something old drop. At this moment of crisis they will go one way or the other: drop the old and grasp for the new, or hold onto the old and lose their chance at the new. The choice may end in triumph or tragedy, grief or joy, but there is always the choice.
The function of the beginning to lead the protagonist towards that moment of choice and to prove to the reader that the protagonist truly loves what they have in their hand and truly desires the thing that is just beyond their reach, that they are actively pursuing through the middle of the book. Until you know what that choice is, you don't have a beginning. Once you know what that choice is, you have a beginning and a middle and two choices for the end. (Though, actually, the nature of the end is usually inherent in the nature of the choice.)
The world you have created and the events you have devised may be workable as they are, but to move on, you need to identify the fundamental moral choice that lies at the heart of your story. (In Hunger Games, Katniss must decide if she is willing to kill to win.) And once you have identified that choice, you will need to go back to your opening and make sure that every word and scene is establishing what your character loves and wants so that you can bring them to that moment of choice that will be the heart of your story.
BTW, if the function of the beginning is to establish values, and the function of the middle is to lead the character to the choice of values, the function of the end is to prove, thought the subsequent action, that the choice has been made.
It becomes clear from the comments that I have not succeeded in getting my point across to all readers. Some took my answer as a refutation of discovery writing, as a statement that you have to work out the entire plot in detail before you being. Nothing could be further from my intent.
Jonathan Karl cites Stephen King's analogy of storytelling to unearthing a fossil in which King describes plotting as a jackhammer that will likely destroy what you are trying to unearth. This idea that the storyteller is discovering rather than inventing their story is very common in what storytellers write about their craft. To me it makes sense given the commonly accepted notion that stories are something very fundamental to the human psyche. (Personally, I go one step further, believing that stories are fundamental to language itself.)
But if a story is a fossil you are uncovering, it follows that the first thing that you find sticking out of the ground that looks like a bone may or may not be part of a whole story. It might just be an incident or a vignette. You may have to dig a little to figure out if you have found a dinosaur or a dead stick.
When I say that if you don't have a middle and an end, you don't have a beginning, I mean that until you unearth enough of the fossil you don't know if you have a dinosaur or a dead stick. What you think may be a beginning may turn out to just be an incident or a vignette. Beginnings, by nature, begin something, and if there is no middle and no end, then what you have is not a beginning.
The question is, how do you tell, once you have dug up what you think might be a beginning, if it is actually a beginning or not? King talks about starting a story by putting characters in a predicament. Does the incident you have uncovered put your characters in a predicament? If it does, then your characters will naturally try to get out of their predicament and you can write the story (as King describes) by watching them try. Alternatively, if you are a plotter, you can work out a way for them to get out of the predicament and then write it down. Whichever method you use, though, there has to be a predicament, and if there is no predicament, there is no story and thus your incident is just and incident and not a beginning.
What is the nature of a predicament that is capable of driving a story? My contention is that all story predicaments are at their heart moral. They are a choice between values. A problem that can be solved by mere mechanical ingenuity is not a sufficient predicament to drive a story. There has to be a moral element, a choice between values, a sacrifice required to gain a greater good.
If your beginning has not established that moral predicament, or at very least established a set of values in which an eventual predicament can clearly be seen, then it is not a beginning.
Inherent in the discovery writing process is that sometimes the thing you think you are discovering will turn out not to be a story. If you are a discovery writer, you have to learn to recognize when this happens and move on to try to discover a story somewhere else.
If you are a plotter, on the other hand, you have to realize that a plot is more than a sequence of incidents. It is a device for bringing a moral predicament to a head, to a moment of decision. If you are going to plan, this is what you have to plan for.