"Suppose" that in a train, the acceleration of the train is great and the carriage is closed. Can people fall at the origin after jumping off the train? How to prove this problem?

This is a basic problem of relative motion, but it is not simple. This is because acceleration, in theory, the magnitude of acceleration is not important. Even if the acceleration is small, as long as there is, it is no different from nothing. Whether there is acceleration or not is essentially the difference between inertial system and non-inertial system.

(1) If there is no acceleration, it is an inertial system, and the calculation of relative motion is very simple, so Newton's law of motion can be directly used. For example, if there is no acceleration, then:

From the train, people can move freely in the carriage, just like on the ground. After jumping, they can naturally fall back to the original point. People can't feel the train moving, nor is it moving horizontally, but moving up and down.

From the ground, it's the same: the train is walking, and people are walking at the same speed as the train. People jump up with upward displacement and speed, but they still keep the original speed in the horizontal direction-because of inertia. After people fall, the train also moves to the same position, that is, it still lands at the "origin". From the ground, people's trajectory is actually a parabola, while the train's trajectory is a straight line. People still don't move horizontally relative to the train when they are combined in pairs.

(2) If there is acceleration, which is your problem, then the train is a non-inertial system. If the train is still taken as the frame of reference, Newton's laws of motion cannot be directly used.

Regardless of theory, just from experience: If the train speeds up (or slows down), can the people on board feel nothing? Can you still move freely on the train as you do on the ground? People will inevitably feel the force given by the floor or the seat (in the horizontal direction), and I am afraid that I will stand unsteadily without holding it. Therefore, the results are likely to be different.

The train is a non-inertial system, so Newton's law of motion cannot be directly used; But the ground is still in the inertial system, so we can observe that the train moves forward with acceleration a and people jump from the ground. Needless to say, in the vertical direction, in the horizontal direction, people will advance at a speed because of inertia: that is, the speed when the train takes off (assuming U). After people jump off the floor, they can no longer bear the force of the train, and the horizontal speed will not change (the air influence can be ignored). So from the ground, the trajectory of human movement is still a parabola. Let's look at the train again: after people jump, the train still accelerates, so the horizontal speed of the train behind is higher than that of people (if the time of people in the air is t, then after people fall back to the ground, the speed of the train becomes: v=u+at). During the whole process, the lateral speed of people remains the same, while the speed of the train increases, so the distance traveled must be different. Compared with the above (1), the track of the train is still a straight line segment, but its length is longer than the horizontal span of the corresponding parabolic segment. Therefore, when people retreat, they can't step on the original position.

Of course, the above reasoning is based on Newton's classical mechanics theory. According to relativity, the algorithm is slightly different, but the result is the same. For your question, the error between relativity and Newton's theory is very small, which can be completely ignored, and there is no qualitative difference.