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I don't understand what "at the start" is supposed to mean for an event that lasts zero time.
I don't think you understand how probability works.
https://outsidetheasylum.blog/understanding-subjective-probabilities/
Ok now I'm confused about something. How can it be the case that an instantaneous perpendicular burn adds to the craft's speed, but a constant burn just makes it go in a circle with no change in speed?
...Are you just trying to point out that thrusting in opposite directions will cancel out? That seems obvious, and irrelevant. My post and all the subsequent discussion are assuming burns of epsilon duration.
I don't understand how that can be true? Vector addition is associative; it can't be the case that adding many small vectors behaves differently from adding a single large vector equal to the small vectors' sum. Throwing one rock off the side of the ship followed by another rock has to do the same thing to the ship's trajectory as throwing both rocks at the same time.
How is that relevant? In the limit where the retrograde thrust is infinitesimally small, it also does not increase the length of the main vector it is added to. Negligibly small thrust results in negligibly small change in velocity, regardless of its direction.
Unfortunately I already came across that paradox a day or two ago on Stack Exchange. It's a good one though!
Yeah, my numerical skill is poor, so I try to understand things via visualization and analogies. It's more reliable in some cases, less in others.
when the thrust is at 90 degrees to the trajectory, the rocket's speed is unaffected by the thrusting, and it comes out of the gravity well at the same speed as it came in.
That's not accurate; when you add two vectors at 90 degrees, the resulting vector has a higher magnitude than either. The rocket will be accelerated to a faster speed.
I don't think so. The difference in the gravitational field between the bottom point of the swing arc and the top is negligible. The swing isn't an isolated system, so you're able to transmit force to the bar as you move around.
There's a common explanation you'll find online of how swings work by you changing the height of your center of mass, which is wrong, since it would imply that a swing with rigid bars wouldn't work. But they do.
The actual explanation seems to be something to do with changing your angular momentum at specific points by rotating your body.
If we figure out how to build GAI, we could build several with different priors, release them into the universe, and see which ones do better. If we give them all the same metric to optimize, they will all agree on which of them did better, thus determining one prior that is the best one to have for this universe.