Schrödinger's hat

And I apologise if I appeared arrogant. I have no intention of skipping the mathy stuff, but I like to give it as much context as I can before I get started. I was under the impression that I had a vague understanding of the idea of a tangent bundle, but this could easily be mistaken. Last year I did an algebra course in which I did something with Lie algebras, locally flat spacetime and proving some properties of Casimir invariants and something a lot like the Bianchi identity -- among other things which were all presented extremely abstractly. I managed to do fairly well in the course, but I have absolutely no idea what any of it meant and I remember very little of it. I was trying to ground some intuition about what [math]G[/math] is in terms of the more familiar wave equations in hopes of bridging some of the gap from the direction of what I am familiar with -- rather than struggling through the abstraction until I realise the gap had been filled. I also realise that thinking in terms of extrinsic curvature is not very useful and can be quite misleading, but I feel seeing the equation written as (operator)metric=T would help me pin-point the flaws in my reasoning, see why the metric is not like a potential and move forward. If arranging it in this form is not as simple as I thought then say so and I shall stop annoying you.

Yeah, I'm starting to get a handle on ideas of connections and tangent bundles, but I feel like this is an important step in bridging the way I currently intuitively think of things in terms of potentials, fields and curvature of things embedded in a frame that I have to make before I can get a handle on intrinsic curvature. To clarify, I can handle the idea of [math]g[/math], and what it taking on different values means for spacetime and events within, but I'm having trouble thinking about [math]G[/math]. So far it feels like curvature of [math]g[/math], but in trying to expand things in terms of Christoffel symbols I lose the plot. In addition there's confusion from it both being used as the metric/operator for mathematical tricks, and as the thing on which we're operating with the Riemann tensor. I can see curvature (two dervatives of the metric) is one way of thinking how the Ricci tensor comes into it, but I cannot fathom a way for the scalar to come into it other than 'because that's what makes conservation of matter work'. I suppose I could be a bit naive in trying to think of this in terms of 'things' and 'operators' but this is the only way I can see of moving forward.

I'm trying to wrap my head around GR, and one thing I've noticed with every EM or quantum eqn I've encountered can be written in the form: (curvature+a bit)Potential/vector potential=source (except dirac, although dirac solutions are a subset of solutions to one like it) Also reformulating newtonian gravity to be SR invariant gives an equation very much like Maxwell's equation (ie. the weak field approximations up to a couple of constants I couldn't track down). Ignoring cosmological constant for the moment. Does anyone know if Einstein's field equations can be written in a form a bit like (I apologise for mangling notation) [math] [\nabla_\mu,\nabla_nu]_{\alpha\beta} g^{\beta\alpha}+Ag^{\alpha\beta}=kT^{\alpha\beta} [/math] where [math] A[/math] is an operator involving two derivatives. Is thinking of a metric as a little bit like a generalisation of the idea of vector potential horribly wrong, or is this concept useful?

Totally derivative of my egocentric model!

The Ultimate M, for the level of detail you seem to be after it might be a bit more efficient to go and find a book. I could write out all this stuff, but I would only be reproducing work which has been done by others, and my explanations would probably not be as good as someone who has spent years writing a book specifically on the subject. Here's here's a link which may be useful (External links on wiki can be a great place to look for these things) http://craig.backfire.ca/pages/autos/horsepower Read that and come back with any questions you still have. If it's not detailed enough for you I'll try and find some book titles.

My ubuntu installation is currently in need of cleaning up and upgrading. I've used: Octave, maxima, matlab (not free ), and tried a host of others on there. The only simulation software I could find was quite cumbersome to use and not really any more efficient than doing things in matlab. Any suggestions for specific packages?

To clarify what mississippichem said in case of confusion, the quadratic formula as he posted it applies to the first form I mentioned, when trying to find f(x)=y=0 or: [math]ax^2+bx+c=0[/math] Note that this isn't the same as the form khaled posted, which is much less common.

One can fairly safely assume that the gearbox will be optimized for speeds between 60 and 110km/h with those power figures -- unless the bike is very old. Five-six speed gearboxes are normal and even four speeds is enough to get a decent range of ratios. Even cruisers and scooters behave fairly similarly at slow speeds (up to 100km/h) with modern suspension on good roads, learning to control them is a bit different, but the performance is much the same.

Compressed air is viable and currently being commercialised. Look up the Tata. Well, seeing as you're being more generous with the type of vehicle used let's run some back of envelope equations again. CaptainPanic already showed a flywheel is borderline viable at high speed. It would work quite well at lower speeds, especially as this avoids the gyroscopic problem to an extent. Let's look at spring power. A very efficient 50cc scooter can go about 100 miles on 2-3 litres, or 25MJ at the back wheel. You could probably halve that, or better if you were willing to do the journey at 30km/h Springs produce around 0.0003MJ/kg so being generous, that would mean about 3000-8000kg of spring. This is a lot more than a scooter's 50kg. If you were willing to go a little slower, and build something like then you might do a bit better.The latest record is 10,000 miles per gallon, or ~2,500 miles per litre. Even highly efficient engines will still be around 50% efficient, most of the gains are from reducing friction/air resistance. That's 2MJ per 100 miles, or ~700kg of spring. This means that it's probably within our technology to do 100 miles (or at least 10 miles) on a very flat track with a spring powered car, but it would be a prohibitively expensive engineering project. We could also cheat and use the not-quite-available-yet-but-soon-we-promise win button for all engineering projects. If this is true, long carbon nanotubes would be roughly on par with batteries as an energy store, and cars capable of travelling 100miles at incredibly high performance (they do not suffer from power draw limits like batteries) would be possible with this single additional technology.

Tx swansont, but I meant I couldn't parse his sentence.

I've seen different conventions in different textbooks, the most common: [math]y=f(x)=ax^2+bx+c[/math] Another (wiki calls this standard form) is: [math]y=f(x)=a(x-x_0)^2+y_0[/math] This allows you to quickly draw the parabola that it represents as you just get a standard parabola [math]y=x^2[/math], scale it in the y direction by a factor of a, then place the turning point at [math](y_0,x_0)[/math]

Torque is a measure of how strong the rotation is. Roughly equivalent to force. 14Nm means that, if you were to put a 1 metre wheel on the crank shaft, you would have to put 14 Newtons of force on it to slow it down. Power is force multiplied by velocity, or torque multiplied by angular velocity (rate of rotation). If you had a perfect and infinite range gearbox the only thing that would matter would be peak power, you could just rev the engine at 7500rpm all the time and get maximum acceleration. Unfortunately there are limitations to gearboxes, engineers have to trade off weight, life time, strength and other factors, so you have a limited (and usually discrete, although many scooters now have continuous) range. This means the amount of torque is important. High torque also usually comes with a big flywheel, and massive pistons. This helps smooth out the power over the course of one stroke a bit. It means the engine won't lose revs quite as easily under load. Now, on to torque and power curves. The higher the torque, the higher the power. But where peak torque occurs is important too, along with how quickly it increases. Here is an example of a highly tuned comparatively small engine. It produces a moderate amount of torque but revs quite high. This means it produces lot of power, but only at high revs. It will go just as fast as something with the same amount of power, but it will require changing gears more frequently, and will probably not accelerate as quickly at low speed (depending on what ratios are available). This doesn't really matter for an engine this powerful, because the motorbike is limited by the amount of force that will take the front wheel of the ground and the rider can do things like slip the clutch. This kind of torque/power curve is known as being peaky (at least compared to my other examples). Small, highly tuned engines tend to act like this, in extreme cases (single cylinder racing two strokes) this behavior is known as a power band where both the power and torque increase dramatically for a short rev range and then drop off again. Note that the peak torque (the light line) and peak power (the heavy line) are quite close in revs, and the peak torque is not far from the maximum revs. These are ways that you can detect a highly tuned engine that will act in this way. Here is an example of two engines, both of which are extremely powerful for a motorcycle. One of them has a moderate or somewhat flat torque curve, the other has a very flat (even decreasing) torque curve. The Vmax is a balance between getting the largest possible amount of power out of an engine, and making that power available without changing gears/waiting for speed to build up, this will also make the bike accelerate more quickly in general compared to a peakier engine with similar power. By increasing the mass of the engine slightly, and detuning resonances so they have broader frequency ranges the torque profile can be flattened. Peak torque is somewhat below peak power. The rocket 3 is an extreme example of peak torque at low revs. It is at half the revs of peak power, and about a third of the revs the engine is capable of. This behaviour leads to almost flat power throughout the entire rev range, changing gears on this motorcycle is only necessary when slow manoeuvring. This is known as a very broad power profile/power band, or a high torque engine. You will often hear the word grunt to describe this, depending on where you live. This doesn't mean the motorcycle will go any faster than one with similar peak power, but it will accelerate at any speed, in any gear (providing the engine does not rev out) about as quickly as a more highly tuned engine will in optimal conditions. The figures you quoted are much more modest, as befits a learner motorcycle. I'm guessing this is either a 250cc twin, 125-200cc single, or something of older design. There is a reasonable range between peak torque and peak power, this indicates that you won't have to work constantly changing gears to find a spot where the engine has power. The overall output of the engine is modest, but enough for around town, depending on aerodynamics and weight I'd say it has a top speed of 110-140km/h. Taking it into hilly areas with >100km/h speed limits may be taxing, but short trips on the motorway should be okay. Without knowing how heavy it is and how many cylinders it has this is less certain, but I would imagine that this motorbike will accelerate about as well as most family cars up to about 60km/h. If you told me how many cylinders it has and what it revs out to I could help more (or just the model number/name).

Things like logarithmic spirals and exponentials or trig functions (which are also secretly exponentials) come up a lot whenever you have something that changes based on a function of itself, such as size of a new shell segment being proportional to the size of the total shell. This ties back to differential equations and self-organising structures. It's one of many complicated looking structures that come from very simple rules. The exponential function comes up as a solution to almost all of the equations in physics, so in a way I guess you could say that quantum physics is similar. I guess if one were to use polar coordinates rather than Cartesian to display things then many of our graphs would be logarithmic spirals. Let's take -- as a random example -- nuclear decay The rate of decay depends on how much material you have so: [maths]A'=kA[/maths] The solution of which looks like this: Or in polars They also come up a lot in something called a phase portrait, if you plot an oscillating decaying function against its rate of change you will usually get a logarithmic spiral, or something very similar. These tend to be useful for analysing chaotic systems among other things

Well first of all get the functions of x on LHS otherwise you can't integrate (cannot integrate x w/ respect to t unless you know what it is): [math]\int\frac{1}{Rx\left(1-\frac{x}{K}\right)}\frac{dx}{dt}dt=\int 1 dt[/math] Then it looks like partial fractions will work so find A, B st. [math]\int\frac{1}{Rx\left(1-\frac{x}{K}\right)}dt=\frac{1}{R}\int \frac{A}{x}+\frac{B}{1-\frac{x}{K}} dt[/math] Then you can integrate term by term. If you need help/explanation for how to find A and B give me another bell.

Well, since no-one else can help, may as well share what I've found during my search. Sage Looks like a simple and quick way to generate 2d plots/animations. It also has some 3d and interactive abilities, although they do not seem easy to make accessible outside of a notebook. Paraview looks time consuming but incredibly powerful vpython looks to have some uses I'm interested to note that my search keeps leading me back to python projects. Although I cannot rule out bias, this was in no way intentional on my part. I haven't had much python experience, but I'm starting to see why so many people are python zealots. The only thing I am missing is quick and easy web 2.0 style stuff that can be embedded in forums/blogs. Edit: Pymol looks pretty nifty if you're a chemist This is almost perfect for quick and dirty calculations/animations. Easy creation of parametric surfaces with time. Big let down is no scripting/find and replace/bulk editing of parameters so if you want lots of lines you have to edit them all by hand. Also no simple way to save as a gif (probably patent issues &*#$ compuserve), but that'd just be icing. Also mathematica keeps popping up as being the best/most recommended for this kind of thing. Anyone have any comments? I found matlab to be fairly efficient if you spent the time to write lots of scripts, but setting things up took some time. Is mathematica useful for small projects?

One option that no-one has mentioned so far is heat. Get the room (especially the surfaces, but hot air will do this indirectly) warm enough and the water will not condense. This is generally the approach used in cold areas in which I've lived for things like stopping condensation on car windows in the absence of air conditioning. A combination of hot air coming in and ventilation will also carry water out of the area more efficiently, although this is hardly energy efficient and if your country does not have an excess of hydroelectric/other renewable power then it is hard to justify. Fan heaters pulling air into the room and fans blowing the air out works very well. Additionally this does not make the area as unpleasant for sport/etc as one would think. The lower relative humidity levels make sweating more effective as well as making it less muggy.

Neat, did you have any query/question or are you just telling us about it?

The arrangement of the spheres is important. The potential of each charge depends on where all the other charges are, not just the ones in one sphere. So the charges in the inner sphere actually have higher potential until they are on the outside sphere (the region with lowest potential which the charges to get to). The charges will flow until they all reach the lowest potential they can get to, at which point they are as far away from each other as they can get -- on the outside of the sphere. Also please don't double post. If we see one thread we'll likely soon see the other.

I suppose if you were super-good at folding proteins you could create some kind of prion which was based on a protein only a few people had. Even assuming such a prion/protein exists for a given protein expressed by your target, it would either have to be damaging once it caused the target protein to fold, or the target protein would have to be essential (ie. it might just turn a brown haired person grey). On top of that, finding it would be far harder than curing cancer/aids etc. Other options would be making something that will catalyse/react with the expressed protein, or somehow hijacking the immune system. Not sure how plausible that would be.

I don't see why this is worth getting so worked up about, there's only one vaguely real/consistent thing, which is momentum-energy. Ambiguity in these terms clearly came from somewhere, otherwise there would be no argument. I think the most useful thing to do at this point is start thinking of what you can do whenever you talk about these concepts to reduce this. There are three useful ways I know of to break momentum-energy down. 1) The scalar product of momentum-energy of a system/object of interest. I don't think we usually care why it's there, just how much there is. 1 photon has none of this, 2 photons can have some if they aren't in the same direction. [math] \left| \sum_i P_i \right| [/math] 2) The scalar product of momentum energy of a single elementary particle, or the sum of these quantities for each individual particle. This is the only time that rest mass is truly meaningful and useful, although for composite systems rest mass is a good approximation. Any time you're dealing with anything as or more complicated than a proton you don't actually have rest mass. [math] \sum_i \left| P_i \right| [/math] 3) Time component of momentum-energy in a given frame. [math]\sum_i \gamma_0\cdot P_i [/math] Edit: I suppose transverse mass would be a fourth. Quite useful, although it seems like a silly concept to call mass in any reasonable sense. The only issue comes up because we used to call them all the same thing. If you put something on a set of scales you're always measuring 1, which is degenerate with 3 because you have to hold it still If you're using non lorentz invariant equations you're using/measuring 3 If you're doing theoretical work, or particle or nuclear physics you will probably use 2, or sometimes 1. When 2 is appropriate, it is usually degenerate with 1 They all get called mass for a number of reasons I can see: People are lazy and just shorten relativistic mass to mass People are lazy and just shorten rest mass to mass People aren't using/don't care about lorentz invariance They decided to redefine mass as one of them, but in no relativity text I have seen -- caveat: I do not read many papers -- do people seem to distinguish between 1, 2 and 3 correctly, they either call 1 and 3 relativistic mass and 2 rest mass, or 1 and 2 rest mass, and 3 relativistic mass. Sometimes energy/relativistic energy is thrown in here too, to add confusion. Suggested solution: 3 shall be known as total energy/relativistic energy -- could do with something better here 2 shall be known as rest mass, most situations in which it is useful/will be used it is degenerate with 1 1 shall be known as mass or rest energy These definitions are as close to consistent with everything I have read as I can manage without inventing new words As the term relativistic mass is used for both 1 and 3 very frequently it should be avoided because it causes confusion and ambiguity. Further thoughts/edit: How often is force a useful concept in a relativistic setting? This seems a bit like trying to keep one foot in each pond. Again, I'm inexperienced, but surely a (2?)-form/rotor of some kind is much more useful than something projected onto a frame, at which point you'd be using a vector for momentum anyway? Oop: Didn't read all of swansont's post. Invariant mass==1

Have you thought about a coil gun or rail gun? A solenoid may be a simple way to use the same concept, although you'll be accelerating more mass.

I am after some free/cheap visualisation and animation tools for explaining mathematics/physics and general science concepts. Something/things that can fairly quickly and easily make 2d and 3d plots, add text and drawings to them. Basic physics simulation/animation tools would be great, too. I have tried this in the past with matlab/octave plotting tools and getting decent animations tends to be extremely time consuming, as is drawing things in general tools like CAD. Can anyone suggest tools, or ways of speeding up the process? Which skills would you recommend acquiring (ie. is 3d modelling/someone's favourite programming language worth the effort?) Tools for other sciences are welcome too, this may be of interest to others

I don't know about you, but all my Phillips-head screwdrivers are globular clusters.