Johnny5

How do you come up with that formula? I believe I saw something resembling it at a NASA site' date=' I'm gonna check to see if i can find it again. The article was on the coriolis force. I actually found it, its for centrifugal force, not coriolis force, here it is NASA: Derivation of centrifugal force Look down and you will see this: (d /dt)i = (d /dt)r+ w x That is Tom's expression practically. How in the world do you come up with it is my question? Let both operators operate on vector A, and you get: dA /dt = (dA /dt)rot+ w x A So the LHS is the time derivative of vector A, in as seen from some external inertial frame of reference, and there is an equivalence. So I guess what I am asking for, is to see a proof of the equivalence. I read the article at NASA before, and didn't know where they came up with the formula above. I'm going to read it again.

If you were going to teach someone about physics involving rotating frames of reference, where would you begin? In other words where is the best place to begin? I was thinking about starting off with the definition of angular momentum, and going from there. L=R X P From there you could go on to cover torque, and moments of inertia. The problem is I'm not sure the best place to start. The moment of inertia tensor is not for a beginner, so I need someplace else. Any thoughts? PS: My goal is to understand gyroscopes in particular, with a view towards quaternion algebra, and how it is used to avoid gimbal lock. Also, I want to tie the work here, to SO(3), which I am discussing with Tom Mattson in another thread. There is just so much about rotating frames of reference out there, it seems difficult to compile it all, and make sense out of it, but that's what I am trying to do. So any suggestions at all would be most welcome. Thanks

What are the axioms of a group? I suspect I know them, but i've read conflicting answers from time to time.

Nietzsche pondered it. Time had a beginning, regardless of this theory, it wouldn't matter what cycle we were in now. In order to even attempt to reason about time, and draw conclusions with certainty, one needs to start off with statements which are true, and known to be true. How would anyone know there was a first moment in time? I would suggest simply build up an axiomatic theory of time, and use that as an axiom. It is possible to construct an argument in which you deduce that time had a beginning, but it's somewhat contrived. Also, Stephen Hawking recently tried to prove that time had a beginning using umm, oh yeah... using the concept of imaginary time, but the problem with that boils down to using square root of -1. No matter how you look at it, the negation of the fact that time had a beginning, leads to a multiplicity of contradictions. But you need a highly sophisticated pre-developed, axiomatic theory of time. Regards PS: Here is one idea i had, but eventually decided the argument was too complex to carry out mathematically... Imagine everything running in reverse, from right now. Visualize the planets reversing their spins, and moving backwards along their orbits, broken things coming back together, go back more and more. If we take current macroscopic astronomical observations seriously, as we go more and more back into the past, relative to now, things will be getting closer and closer together. Perhaps a computer could be used to project how far back in time all of the material was concentrated into one mass. Now comes the idea, but as I said it's too complex to prove useful. Eventually, there would come a condition of the universe, at which the reversal of motion would cause some problem. Like, for example, the temperature of the supermass approaches infinity if there was no first moment in time. Something along these lines. But you aren't going to deduce the answer this way, for a lot of reasons. But my point is, time did have a beginning, how you argue that fact is up to you.

How did you get that formula?

But there is a difference in the physics. But see i know you know that, as well as I do. Also, i began reading an article on Geometric algebra, that seems related somehow. Kind regards

Thanks Tom PS: Rotating the axes isn't the same as rotating the vector, but I like what you did here. I'm also going to have a look at this quaternion thing, i think thats a goofy name but whatever, ummm in the thread above, with Diana Gruber and someone else having a semi-friendly discussion, eventually Diana quoted some formulas obtained by Gibbs, which i am going to have a look at, but from what I gathered, the main thing about Gibbs' approach was that he avoided using the square root of negative one. The thread was on the whole... very illuminating. Also, there was mention of Gimbal lock. Regards

Ok. "The matrices you cited are in fact for rotating the axes counterclockwise by an angle A." Hmmmm

I found this, and was wondering about it. Id like to hear some thoughts on this. Progammer lashes out at quaternions From what I have gathered, a computer programmer, Diana Gruber, wrote an article about how terrible it is to use quaternion algebra to develop the rotation matrices. What made the whole thing notable, was because she works for some 3D gaming site, or something along those lines. I read through her article, which was subsequently lambasted in the post at the link above, and in her article, she actually goes through extensive effort, using classical trignometry, to develop rotation formulae. But, then I read the post at the link above, and the person there mentions SO(3), here:

Let me see if i understand you right. Are you saying that if there is an inertial frame somewhere, a frame in which Newton's law of inertia is true, then using the formulas from GR, the conclusion is that space is flat? Do I have that right? Regards

Ok, I see that now. In your initial post, you said rotate the vector, and then later you said, rotate the axes. they are not the same problem. No problem Dave. This is quite subtle Tom. Ok I'm gonna draw a new diagram, and see if i can get your answer. I have a question. Is A necessarily positive? Orientation Methods We have vector <x,y> in unprimed frame S. The same vector is <x`,y`> in frame S` Frame S` has the same origin as frame S, but is rotated slightly by angle A. Now, if A=0, then S=S`, but it is not the case that A=0. I made a drawing, and I have A = 30 degrees, and the vector V is in the first quadrant of S, at about 80 degrees to the x axis of S, and 50 degrees from the x` axis of S`.

It is an issue with complex numbers. Let me go back and show you why. Now, some have a problem with the complex numbers, and others do not, but regardless, it has been highly developed over the past few centuries. So here is what i did, during the solution of the problem. First, I started with the information given, as true. You did as well: Up till then, i didn't already know that the roots of that polynomial, were necessarily complex. But then at this point i did. I looked back at your question, and didn't see any reason why r couldn't be a complex number. So that finished off the case where (1+r+r2)=0. Then I covered the mutally exclusive case, where the polynomial above is not equal to zero. In this case you can divide through it, and then you can rapidly see that r=2. The two cases above are mutually exclusive, and collectively exhaustive, so i was done. To summarize: Unless there is a reason why (0,0,0,0,0..) cannot be a sequence, or unless there is a reason why r cannot be a complex number, I don't see anything wrong with a comprehensive answer. There was nothing at all wrong with what you did. You just asked to see alternative approaches to answering the question, so i tried to see how I would solve it. I happen to like the geometric series, which is where i encountered arithmetic/geometric progressions in the first place, and this was many years ago. Kind regards

Alright, this is a place to start. Rotation Matrix Rz ( F ) You start with a three dimensional rectangular coordinate system, and x axis, a y axis, and a z axis, meeting at the origin (0,0,0). Let V(x,y,z)=<x,y,z>=xi+yj+zk denote an arbitrary position vector in the frame. Now, we want to "rotate that vector about the z axis, by 90 degrees." I'm not sure how to visualize this. My question is how do you come up with the rotation matrix in the first place. I don't see it just yet. I understand the last part though. Once I have Rz ( F ), i let phi =90 to get: Rz ( 90 ) Which is the matrix you gave. Then matrix multiplication gives the answer to the question. Rz ( 90 ) V = answer

Ok, thank you. Can you show me a basic example of something, which forces me to use 90% of the ideas here? PS: Just so you know, the reason I asked the question, is because I am trying to understand gyroscopes, precession, nutation, spin, roll, pitch, yaw, Euler angles, that kind of thing. In the process of trying to understand gyroscopic motion, I came across SO(3), and it seemed connected to most of the things i just listed, as well as something called quaternions, developed by William Rowan Hamilton. So I am trying to learn all these new things, just to understand how a gyroscope works. If I could understand just this one thing, I would probably understand the whole of classical mechanics, at least that is how it is beginning to look to me. Another thing of possible interest, is that all of this seems related to the Sagnac effect, which Geistkiesel and Tom Swanson and I were discussing in some other thread. I now realize that mechanical gyroscopes can be replaced with optical ones, which are based upon the "Sagnac effect." And Dr. Swanson also mentioned, that the linear sagnac effect was somehow related to the Michelson Morely experiment. So I can see some kind of synthesis that can happen, if I learn the right things. I expect to use linear algebra to understand the gyroscope, possibly quaternions, possibly Euler angles, and all of that stuff seems centered on SO(3).

Can you be more precise?

Hmm i need a moment. let me answer the previous thing, and I will come back and think about this.

I want to solve a problem, to help me recall some more linear algebra. I want to solve a matrix system, and I want the answer to be a straight line in three dimensional space. So intuitively, i know this: If two planes are parallel they have absolutely no points in common, but If they are not parallel then they do have points in common, and the set of all points they have have in common lie on one and only one infinite straight line. Now, i know that the form of an equation for a plane in three dimensional space is: Ax+By+Cz=0 So choose two planes from the set of planes, but make sure they are not parallel. At this point, i want to solve a system of simultaneous linear equations, and write my answer in the form of an arbitrary straight line in 3D space. This is the part I cannot remember how to do. Thank you

Question: What is so special about the elements of O(3) whose determinant is +1? Also, what does SO(3) have to do with rotations? Regards

I thought his keyboard was broken.