Project management is simple - based on the information you have, predict
- when the project will be done (schedule),
- what the project will cost(Cost) and;
- how well it will fill the need (quality).
If I understand the problem, you're doing more than your fair share of the work. There are two possibilities:
"I can barely resist not doing the [w]hole work of the team by myself " - in which case the cost of the project is measured in the impact on your schoolwork and on your future earnings potential. (Cost can be opportunity cost as well as dollar/pound/franc/etc. cost). If the project delays your degree it will have a huge impact on your lifetime earning potential.
You can't help doing the work because the rest of your team is not motivated to do the work. There are many things you can do from a leadership perspective, but from a project management perspective, you just adjust the schedule, cost and quality based on observed behavior. (This is a situation in which I find myself frequently). "You agreed to do X by date Y, but you're only 50% done, which means that X will not complete until Z; that will delay the completion date for entire project until date W." If everyone is comfortable with W, then fine. But if W creeps forward until W is long after graduation date, then that discussion may lead your team to either change behavior or close the project.
Just saw the update to the question, in which you point out that one of the potential costs of the project is your friendship. Excellent observation, and you need to find a way to discuss and quantify that potential cost. (realistically, there is no way to quantify the value of friendship; in the past in similar situations what I've done is to give everyone veto power. If anyone feels like this project's burden is rising to the point where it threatens the friendship, then that individual can pull the plug. Others agree that there will be no judgement, no condemnation. I've worked on projects where the stress was sufficient to require medical care. Is this project really worth that cost?)
Emphasis note that this is not judgmental, this is just evidence. I don't care why you missed the deadline(illness, death in the family, alien invasion), I just care about the impact on the project. PM isn't about judging people, it is only about the impact on the project.
In both cases, the goal is to measure the impact on the project and then plan mitigations. Your project is at risk of failure, and you need to mitigate that risk. Analyzing the project permits you to ask over and over again,
is this project worth it? Is the benefit of this project worth the cost? Or do I need to reduce the cost, change the schedule, or change the scope (or quality)
I missed the fact that this is schoolwork (I thought it was a side project). It is very difficult to "close" a school project. Back when I went to school, there were no group projects; now they seem to be commonplace. I've got two additional suggestions
Manage scope. If you are not getting team participation, then first make a plan and second talk to the professor. Explain the problem and what you've done, and make sure to indicate that you'll continue to work to motivate the team. But if participation doesn't increase, then you will scope your participation to a clearly identified portion of the project. (picking an arbitrary example) "I will outline European history from 1200 to 1600, but I personally will be responsible for Turkish history 1300 to 1450. I have four courses and I'm not willing to fail those courses in order to pass this course, so I have to establish some limits."
Build a smaller group. My girlfriend had a similar problem in her grad courses; she went to the regular group and worked with them, but she then scheduled a smaller group meeting with the people who were willing to work. The core team knocked things out because they were working with one another. Harsh, but sometimes the only realistic solution.
In truth your problem goes beyond project management - I've tried to limit my answer to project management interventions.