K^2

Relativity did not come about from total failure of the principle of extreme action. In fact, entirety of Classical Mechanics is preserved in General Relativity. The only change is that coordinate system transformations are local in General Relativity, rather than global. That has all kinds of interesting consequences, and it introduces small corrections to some approximations. If you take away conservation of momentum/energy, everything in physics is wrong. This is not an exaggeration. This isn't a matter of "our choice of coordinate system wasn't general enough." It's "every single conclusion we've made has been based on wrong assumptions." Yes, new theory would have to explain old data. But more importantly, it would need to explain how a theory that was completely wrong explained the data we had. Again, we aren't talking about small algebraic corrections due to higher order terms. We are talking about completely wrong logic leading to seemingly correct results.

We already are casting a wide net and looking for alternative models from the ground up. What you are suggesting would result in a total stall, and people being hesitant to invest in future technologies, with good reason. All it would do is stall our economy and tech. We aren't talking just about a few centuries of stagnation. Our civilization can't handle that. We are talking about collapse of modern science with nothing to replace it, which will result in collapse across the board. Sure, we'll probably do alright in the long run. But there isn't an improvement. If we are capable of finding better solutions, we are much better positioned to do so now, with the technology and R&D investments helping along. Finding a flaw in conservation laws would not help us along. So looking for these would, quite certainly, be a net loss. P.S. I don't know if I should mention quasimomentum. It's relevant, but I'm worried that crackpots are going to jump on that. They like to incorporate things they don't understand into explanations for why their "theories" work.

Let me tell you a story. NASA runs many AI projects for Mars rover. A little over a decade ago, they had a test run with very interesting results. They loaded their mock rover with a map of the terrain, they set it down on the ground, and they told it to move. It didn't. It stayed in place. They checked for mechanical obstructions, power to the motors, and found no problems. They reset, and still couldn't get it to move. They then began looking for problems in navigational systems, localization, mapping, and so on. And there was nothing wrong with any of it. Eventually, they discovered the problem. They programmed the map in with the wrong elevation offset. It's a flat offset, so it shouldn't have caused a problem, except, according to rover's localization system, ground was 10 feet bellow. The only place where it new for sure that the ground wasn't 10 feet bellow was right underneath its wheels. So according to its navigation system, move in any direction would result in a dangerous drop, which is something it was explicitly programmed to avoid. And so it did not move. That's what discovering failure in energy/momentum conservation would be like. We'd know precisely one set of safe technology. Stuff we are using right now. Everything else would be a huge chasm. For all we would know, the nukes we've built haven't caused a chain reaction that would destroy the planet only by chance. Primitive tech would probably be pretty safe to experiment with, but anything remotely modern would be absolutely unsafe, because we simply wouldn't have reliable machinery to tell what we should be messing with and what we shouldn't. Energy conservation, and in fact, just general properties of the symmetries in question, guarantee us that we can only screw up so much. That's what we've been relying on to say that it's probably ok to keep pushing the boundaries. Anything we do screw up will have local consequences. Not so if these principles don't hold. The moment we discover a failure like that, we'd have to stop. It will not be just a stall of our science advancing, but all of our tech advancing as well.

Science is a constructive discipline. We are entirely ok with looking for things that are wrong within it, but there must be a goal. For example, there are teams at accelerator facilities looking for heavy photons. They almost certainly won't find them, because they almost certainly do not exist. Our theory says so. But it says so based on certain assumptions that don't absolutely have to be correct. And if heavy photons are found, we need to fix these assumptions and readjust our theory. It would be of great benefit, and we wouldn't have to scrap things we've learned. So all in all, it's worth wasting this effort just on the off chance. Because, occasionally, you do find that you're wrong. CP violations are a good example. On the other hand, if Q-Thrusters or EMDrive are shown to work at thrust/power levels advertised, then physics is wrong. All of it. From very bottom to very top. There is not a shred of modern theory we can preserve against it. All of Quantum Mechanics and all of General Relativity ride on the concept of local symmetry groups, and these two devices violate these local symmetries. It is absolutely beyond salvage. And with these go all of classical mechanics, all of classical gravity, all of classical electrodynamics, all of classical thermodynamics... You get the point. There is absolutely nothing left that doesn't rest on these fundamental symmetries. So we have a situation where two things are evident. First of all, if these devices work, we've made correct computations of everything from gyromagnetic ratios of elementary particles to signals we receive from distant neutron stars based on wrong physics. Lasers, semiconductors, superconductors, and even your GPS, all hinging on the assumption that symmetries hold, we got them by total chance in a world where these symmetries do not hold. Unlikely does not begin to describe it. I do not like to use the word "impossible", but I can think of nothing that's more appropriate. And if, on this impossible chance, it does turn out to be true, what do we get? Complete collapse of all of physics with absolutely nothing to replace it. We'd have to go all the way back to Archimedes. It probably wouldn't require quite 2,000 years to catch up, but it'd be centuries for sure. And until then, what? We keep using these devices that produce thrust out of nothing? Who knows what else they produce out of nothing? If momentum isn't conserved, then why should energy be. And if you don't conserve energy, where is the guarantee that when you try to bring these up to full power they wouldn't start spewing black holes, or something. It's absolutely destructive. If these things work, everything we know is wrong, and we should not continue using these things, because we cannot predict outcomes. At all. So not only are the odds of these devices working essentially zero, but so is their benefit. Why invest into something that is guaranteed to be a net loss no matter what?

Working on it. And I'll need to regroup on the sim. I still have one of the orbital elements being miscalculated.

Depends entirely on the hardware we can get. But it looks like VHF would be entirely doable, which means the setup can be as simple as USB-capable micro controller hooked up to an amp, which is hooked up to an amateur radio with sufficient output power. No mess with tracking necessary. Writing software to run it all would then be quite trivial. That would also mean that the sat just needs a burst VHF transceiver, which can be purchased or built without major expenses.

This is only strictly true during the vertical stage of the ascent. Once you begin gravity turn, you briefly need a bit higher TWR. But that tends to work out, since your rocket is getting lighter.

A field with a zero is also a group. Two groups, actually. I'm not sure where you are going with this whole thing. You keep saying that. Yet you aren't providing any reasons for why it's different. It is not. It is useful as a shorthand in a lot of places, just like sqrt(-1) = i is a useful shorthand. Neither is actually true. Map x^y has an undefined point at (0, 0), and you cannot avoid that. How about every time I needed to write code that doesn't crash because of a numerical singularity? I accept his proof, because it is a proof. A strict mathematical proof. You have not constructed one. You simply keep insisting that identity acting on identity must give identity, when the whole conversation is, "What happens if it is not?" You seriously don't understand why it is silly? qemist used all of the definitions set and has derived a contradiction. You introduce contradiction by bringing in a contradicting assumption. I can prove anything the way you "prove" it. Erm... I'm not sure which Wikipedia you are reading, but the version I'm getting states, quite correctly, that Magma is simply a map S x S -> S with no other requirements. It doesn't even have to have identity elements. But if it does, it can have any number of distinct elements that act as identity on anything other than each other. There are no axioms this can contradict, because there are literally no other axioms.

Sure you can. Every element in a field has a multiplicative inverse, unless it's additive identity. When definitions don't work for one particular element, it doesn't hurt the generalization. We make one special exception and move on. And it's not a problem that exception is itself, either. For example, I can point out that, "There exists a real number, such that any number, except itself, to the power of that number is 1." You can tell me right away what that number is, and the fact that definition includes itself isn't a problem. The problem here is specifically with inverses, as qemist pointed out. I can have a magma with any number of distinct identities. It's not useful, but it's not a contradiction by itself. Contradictions happen when we try to assign some useful properties to the binary operation in question.

Gravity constantly transfers momentum over to the craft. You have to bleed that momentum off. That means either applying a force to a static object (ground) or expelling reaction mass. In other words, you cannot repel from plasma without expelling it. That simply does not work.

It's pretty accurate. I mean, I haven't gone through the numbers, but gravitational lensing isn't a terribly complex problem. They just happened to run it through actual ray tracing with other stuff floating about, rather than do a simple deflection angle against a fixed background image, as people have been doing before. There is no reason to think that it's not accurate. The disk only gets brighter as you get closer to the event horizon. If this was just matter of scale, we wouldn't see a black disk in the center. And if that black disk represents extents of event horizon, gravitational lensing should be quite apparent.

Good catch. And that doesn't even require all inverses to exist. So long as there is a single element with both the left and the right inverses, the identities (and inverses) are identical. It leaves me wondering how much more I need to strip down a group before I can get it to support multiple distinct identities. But it's probably moot. I can certainly strip it down to magma with identities, but then calling second identity -0 is absolutely arbitrary. I concede. There is no good reason to have a -0 notation.

It might be just as natural, but it's not what you want if you want two distinct identities. 1+1 = 0 is quite natural, but not what you are going to go with if you are constructing a number line. The other definition is consistent with two distinct identities and every definition up to that point. It is a valid algebraic structure. Again, you are thinking in context of a group. We are not building a group. We are building a group-like algebraic structure with distinct left and right identities. Argument that if left identity doesn't act as identity on the right identity is about as valid as saying that Reals aren't a field because zero has no inverse. Now, if you can find a self contradiction, that's another matter.

0L + 0R = 0L, naturally. It follows from all the other definition. And no, identities don't have to act as identity on other identities. Not in algebras that do not have commutative property. Having only left/right identity (or both) and having special rules for these is pretty common in math. Consult this table. You should read some texts on abstract algebra and group theory. Indeed.

Just to give an example, consider non-commuting addition. In that case, we might want to introduce a left identity and a right identity. Specifically: 0L + a = a = a + 0R for all a that are not identities. If that is the case, and we want to preserve at least associativity, then we define inverses thusly. a + (-a) = 0R, (-a) + a = 0L. That way, a + b + (- = a. Note that (-(-a)) is not a. This brings up the question, what is 0R + 0L? I can't find any reason not to define it to be 0R. Consequently, 0L = (-0R). This leads to an algebra where -(a + = (- + (-a), which may look odd, but is entirely self-consistent. Finally, as means of simplifying notation, I call 0R simply 0, and 0L = -0.

Addition doesn't have to be a group.

It sort of depends on the algebra. One could conceive an algebra where there is a meaningful difference between 0 and -0. But for fields we usually deal with, qemist's reply covers it. -0 by definition is the number such that 0 + (-0) = 0, and that trivially gives you 0 = -0. In practical situations, -0 arises in two contexts. Leszek covered one. If you are dealing with a tiny negative number it can be rounded to -0. Though, more typical notations for slightly positive and slightly negative numbers are 0+ and 0-. The other situation, where -0 arises quite frequently, is computer science (Ninja'd on that by Nuke). The way real numbers are represented on modern computers is as floating point numbers. Unlike integers*, floating point numbers have one bit reserved for the sign, and that is computed separately from the rest of the operation. So if you, for example, multiply 0 * (-1), you get -0 as the result. It doesn't hurt anything, and typically means nothing. I have, however, seen some hacks that use that extra bit of storage for interesting things. It's worth to keep in mind, however, that one of the reasons floating point is set up this way is because 0 is often treated as "very small number rounded to zero." * Integer math is done completely differently. While the high bit can be considered a sign bit, and for the purposes of comparison it is, it's not used in algebra any differently from the rest of the fields. Instead, entire algebra is done modulo 2n, so there is no difference between a negative number and very large positive number. As a consequence, zero is entirely unique. There is no such thing as integer -0. Edit: Nuke, whether == passes or fails depends on how you compare them. In most situations, compiler will know that these are two floats, and will use floating point comparison. FPU will, indeed, report 0 == -0 as true. But if you compare them in some odd context, you might end up with bitwise comparison of memory that will evaluate to false. That usually means you either have an error in the code or you are doing something intentionally creative, though.

You can always just dip your orbit just bellow atmosphere threshold in KSP. Atmo is thin enough there to maintain altitude using ion/plasma propulsion. Is there a mod that would actually let you store ox from scoops, though? I haven't seen anything like that.

And if you try to apply this to a real wing, you'll watch the simulation fail miserably. The problem is that there are a crap ton of vortex states generated by any air foil, and they come in all shapes and sizes. So if you use a fixed grid FEA, you end up missing separation layer, and generating almost no lift. If you simply check for vortex condition and sub-tesselate your grid, you end up with too many elements to do anything with and your simulation stalls. What people really end up doing is a horrible monstrosity stitched from FEA, particles, and a whole bunch of heuristics. And they still, generally, end up working just for specific kinds of geometries. For a game, it might be sufficient to do laminar flow (inviscid, incompressible) FEA and hacking in a fake separation layer that's going to be treated as a boundary by your FEA. It won't be perfect, and you would never, ever do something like this for a proper engineering simulation, but it might be close enough for a simulation game. Fluid dynamics is just one of these problems, you know? As for FAR, I haven't looked at it too closely, but as far as I know, it's a relatively simple version of blade element with some pre-defined polars for the blade elements. Anything sensible for a game like KSP will be some variation on that, to be honest.

That is true with respect to fixed datum/pivot. But if you do flight dynamics with respect to a fixed pivot, your CoM is accelerating relative to the pivot. That is a huge pain to deal with. OP is on the right track, I think. It's much easier to simulate with respect to center of mass, and just recompute the moment of inertia to account for changes.

Afraid not. Moment of inertia is a functional of mass distribution, and you have mass distribution changing in this problem. You need to recompute the moment of inertia, but there is a shortcut. You can precompute the moment of inertia of an empty rocket and only compute moment of inertia of remaining fuel dynamically. If you assume some simple geometry for your fuel, like that it's always a cylinder, but the height and CoM of that cylinder changes, then it should be a fairly simple task. Then, on each update, you'll find new center of mass, take rocket's precomputed moment of inertia, adjust that by MR² using R = distance from empty rocket's CoM to current CoM. Then do the same for fuel using the dynamically computed moment of inertia and R being distance from fuel CoM to current CoM. Finally, add the two together to get current moment of inertia. You can keep doing this for as many elements as you have on your rocket. For example, if you are simulating a liquid fuel rocket and want to take into account that the fuel and ox tanks are located in different places and deplete at different rates. Finally, keep in mind that MR² is only valid in 2D or if your axis of rotation is somehow fixed. (Which makes it behave as if it was in 2D.) For a general, 3D case, you should be dealing with moment of inertia tensor, and you'll need to use the correct tensor version of the MR² offset. Edit: Ninja'd by paul23 with essentially the same info.

Windows 7, 8, 9.

Yeah, it's just the opposite of easy to handle. It annihilates with itself. I wouldn't think on it too much, though. There is no reason for real Majorana fermions to exist, and quasi Majorana fermions aren't going to be any good for energy storage anyhow.

Absolutely. The way you find a trajectory of an object in GR is by solving for zero acceleration. So effectively, you find a coordinate system in which object follows Newton's First Law, and then if you need the force of gravity, you simply find the fictitious force it'd take to drag the object along that trajectory with respect to your choice of coordinates. It's the same idea as with rotating frames of reference in classical mechanics. So as an interesting side effect, in GR you don't need to treat gravity and rotation separately. You can just pack rotation into your metric, and any centrifugal/Coriolis effects will factor into gravity you get.

These experiments aren't new. And scientific community have replied to them pretty much right away. The problem with the setup is that there is no guarantee that the leading edge of the signal is where it is claimed to be. If you send a Gaussian wave packet, of course you can detect its tail before you predict the center to arrive. And there has not been a single experiment to get FTL communication in a setup where this behavior is controlled, despite many attempts. So effectively, it's the same story as with the "faster than light neutrinos" that have been making waves a while back. If you are measuring the wrong signal from the wrong place, of course you can measure something FTL.