Planets in the Fourth Dimension
johncarlosbaez.wordpress.com
johncarlosbaez.wordpress.com
This fourth dimension observation is cool and likely very useful mathematically, but it's also kinda obvious (at least after playing KSP) that if you subtract the time/gravity element from an orbit, it becomes circular. It's equivalent to saying that if you subtract out the gravity effects of the planet you're orbiting, your orbital speed is constant, which just makes sense intuitively.
Calling this observation a fourth dimension is useful, but perhaps unnecessarily complicated for the simple concept.
Then I remembered that it's still accurate enough to be awesome and stopped worrying about it. :)
http://en.wikipedia.org/wiki/Orbital_station-keeping
But anyway, I still hope that Principia mod (the n-body simulation addon mentioned elsewhere in this thread) will get released. This will really bring KSP into Dwarf Fortress level of enjoyment. Also, you'd get to make some really weird-ass stuff:
http://images.scholarpedia.org/w/images/f/f6/NbodyChoreoGamm...
There weren't powerful enough computers to model full Eisenstein N-Bodies Systems until actually very recently (also its complete over kill for inner-solar system travel). Relativity is really complex math, and even modern computer clusters struggle to model very complex systems.
Yes we sent astronauts to the moon using KSP math.
I'm almost positive that we did not.
While patched conics would have been used very early on for rough mission analysis and design, we had a very good understanding of perturbation theory at the time. Wikipedia tells me that the restricted 3-body problem was also essentially solved in 1917. I've seen very detailed plots of the free return trajectories that the Apollo missions followed, too.
Using patched conics for the earth moon system would have resulted in an error on the scale of lunar escape velocity upon entering the sphere of influence of the Moon.
Free return trajectories can be calculated with patched conics. 3 Body problem was solved, but actually doing all the math involved dynamically was far to complex for mission computers.
You forget that at the time NASA was using IBM System/360's which were struggling to maintain 5 megaFLOPs
For one thing, I'm not convinced that your mental models of what a "mission computer" is and what a "mission computer" would be doing, or when, make sense from an engineering perspective. I'm also not convinced that you know what an orbital perturbation is, nor how they would be used to plan and fly a mission.
To take an example from aviation, we didn't have practical simulations of general viscous fluid flow in the 1960s, but it would be meaningless to say that we "used Bernoulli's principle" to fly across the Pacific.
The Apollo vehicles had an on-board IMU. A trajectory could thus be pre-planned and flown to using feedback control. That trajectory would certainly have been calculated ahead of time. Thus, while there would be no need to have "[done] the math involved dynamically," that by no means implies that patched conics were used while in flight, nor does it imply that they were in any meaningful way used for the final design of the missions' flight paths.
Further, on any deviation from the planned flight path, numerical integration methods would yield results that would be good enough for later correction, again, by using feedback control.
If you still need to believe that "KSP math" is how we got to the Moon, go ahead. You've certainly done nothing to convince anyone else, though.
http://forum.kerbalspaceprogram.com/threads/68502-WIP-Princi... https://github.com/mockingbirdnest/Principia
We tend to operate in the usual 6-dimensional parameter space for everything we do (3 position vectors, 3 speed vectors), but there are equivalent parameter spaces that do the job just as well. In some cases, the alternate spaces are more useful - polar coordinates are the most common example.
The paper quoted in the article found a parameter space where very interesting transformations and symmetries take place.
Also, these days I'm reading Max Tegmark's "Our Mathematical Universe" and, being immersed in that argumentation, the whole "real vs unreal" dichotomy seems a bit transparent. Anyway, having formerly played a bit with computational physics, I'm primarily after what's useful.
Seriously, it's amazing how this game can let you understand space travel and make you want to know the math behind it.
So you're piloting ship A and you want to rendezvous with ship B in a stable orbit. Let's assume that A and B are on identical, basically circular orbits, and A is behind B, so your distances from each other aren't really changing much.
The maneuver you're looking for is.... Point retrograde (i.e. "away from the direction that will take you to where B is now") and burn. In the short run, you increase your distance from B (of course), but you also dropped your orbital altitude and therefore the area of the arc swept between your ship and your center of orbit over time... and by Kepler's second law, you are now orbiting faster (more full revolutions per unit of time). So you'll swing under B and eventually come up "in front" of it; time your burns right, and you come up epsilon distance in front of B; rendezvous successful.
It's totally counter-intuitive to accelerate away from something to approach it, except you're basically on a sphere. :-p
After that point no maneuver was daunting in KSP, just something that was a matter of time and making sure I quick save occasionally.
If you have no gravity the particle will simply move in a straight line, and that's not a motion I would call an orbit.
During an orbit, gravity is of course stronger when you're near the planet, and weaker when you're away from the planet. So they equalize the gravity by subtracting the relative time/gravity differential and calling it a separate dimension.
This leaves a constant gravity force and velocity, making certain calculations and transformations more natural in this model.
e.g. gimbal lock solved by quaternian rotation.
Edit: this makes me wonder what other eliptical orbits might look like. Would the 4-sphere have to rotate about the barycenter to give the effect of this: http://www.polaris.iastate.edu/EveningStar/Unit4/Graphics/Pi...
The only time you can contradict a model in Physics, is if the model conflicts with something observable. Isn't that why we have so many coexisting cosmological theories?
Extending the same notion of there being a time dimension that is less dense the further it is from a gravity well - well, it changes certain aspects of how one might look at the universe, such as, say, why distant galaxies are redshifted, why galaxies rotate evenly throughout their volumes, and why the speed of light is a thing.
Looks like he's still battling against Google using the wrong image for him when you google his name: http://gregegan.customer.netspace.net.au/ESSAYS/GOOGLE/Googl...
Baez has earned more credibility than to be written off with a vulgar thoughtless one-liner. He is a Physics major.
This article about a "weird fourth dimension" that's "like time but not time" certainly seems like horse shit to me.
In fact, coordinate transformations are at the heart of the modern conception of classical mechanics: http://en.wikipedia.org/wiki/Lagrangian_mechanics
Using a configuration space that is a good fit for a specific system is not unusual in math or physics. It's the same thing that astrophysicists do when they work in orbital elements, or that engineers and programmers do when they make calculations in Fourier space. It's worth understanding these concepts even if you're not a physicist.
Configuration spaces are so fundamental and useful that they are introduced in the very first lecture on classical mechanics in Leornard Susskind's excellent series of Stanford physics courses: http://theoreticalminimum.com/courses
(Neal Stephenson's novel Anathem also has a simple introduction to configuration spaces in an appendix, because that's the sort of thing that ends up in a Neal Stephenson novel.)
Anyway, people are better off looking at the paper itself (which is rather sober) than reading Baez's summary, as the two are about the same length. I've noticed there's an unfortunate tendency among too many physicists/mathematicians to focus more on trying to amaze their audience rather than to educate them (in it's worst manifestations, you find them spouting outright falsehoods).
Um, no. They are moving in 3 dimensions and when they are projected down to 2-dimensional space they become ellipses.