steveupson Posted May 17, 2016 Author Posted May 17, 2016 Good luck on making direction invariant. That's a handwave if I ever saw one. Of course it is if you're not willing to do the math. i need help with the math.
Mordred Posted May 17, 2016 Posted May 17, 2016 (edited) That's my point there is no math that will make direction invariant to all observers regardless of coordinates. If I could show that mathematically I would win the Nobel prize. Look at the examples I posted. Different observers will measure different directions How they describe that direction is a coordinate choice. What they compare that direction to is also an arbitrary choice. Arbitrary coordinate and reference points are not invariant. Edited May 17, 2016 by Mordred
steveupson Posted May 17, 2016 Author Posted May 17, 2016 That's my point there is no math that will make direction invariant to all observers regardless of coordinates. If I could show that mathematically I would win the Nobel prize. I think you're beginning to understand the issue. This has now been discussed for a while, with others, before the discussion moved here. We need more horses. We need to find someone who can either graph the function or derive it through some magical method. As I said before, my attempt is nonsensical. More likely than not, I used sin instead of sin2 somewhere, or some such.
Mordred Posted May 17, 2016 Posted May 17, 2016 Well tell you what when you show the mathematical impossibility of describing a direction as being invariant to all observers let me know. You can't even show that as being invariant in the same coordinate system. Observer a measures a ball moving in one direction (relative to himself) another observer will state its direction relative to himself as different. That by itself is inherently variant to the observer Even certain measured properties are variant. One example being the measured energy or temperature. No amount of mathematics will change this. It is inherently part of relativity. Google redshift for details It isn't some arbitrary mathematical trick that defines whether or not something is invariant. We would have loved time to be invariant. However tests of relativity shows that as being false. It certainly would have simplified things if light was variant and time invariant. Nature doesn't care about what we wish though. PS hopefully you don't post these speculations on physicsforum. They will instantly lock the thread at any hint of speculation. Yes I'm a member there as well
steveupson Posted May 17, 2016 Author Posted May 17, 2016 No amount of mathematics will change this. It is inherently part of relativity. It isn't changing anything. That's what is difficult to understand. Rather than expressing direction on a two-dimensional surface using vectors, it is expressed as a quantity in three dimensions. Once you actually see what's going on, your entire concept of direction will be stood on it's head. If you don't want that to happen to you, look away now. Is there any physical law that says whether position or direction is the preferred "reference frame?" Again, if you haven't looked at Daniel Cross' paper on "The physical origin of torque and of the rotational second law" then you really should consider doing so in order to better understand the question. -1
Mordred Posted May 17, 2016 Posted May 17, 2016 (edited) Look your still missing the point. Different observers will measure the direction differently. Its always change relative to a reference. That is not an invariant property. Hence direction is a relationship, in the same manner as distance. For example the direction up is relative to your choice of down. (Normally chosen by which way objects normally free fall) A person on the other side of the planet his up is different than your up direction. How you would describe North on Uranus would be different than on Earth. You can draw any line on a circle and state this is zero degrees regardless of orientation of that circle. Edited May 17, 2016 by Mordred
steveupson Posted May 17, 2016 Author Posted May 17, 2016 Hence direction is a relationship, in the same manner as distance. Yes, that is exactly the point. It's what I've been trying to tell you. In one environment (plane geometry) the length is the scalar while in another environment ( _______ geometry) the direction is the scalar.
Mordred Posted May 17, 2016 Posted May 17, 2016 Oh which geometry is direction a scalar? Go ahead name it.
steveupson Posted May 17, 2016 Author Posted May 17, 2016 (edited) Oh which geometry is direction a scalar? Go ahead name it. Look, it's math, it's either correct or it's incorrect, right? The model presents a description of a function that defines the relationship between a reference and some other direction. Call it what you will. On edit> A while back you mentioned rotational symmetry. Think about what "rotation" is. In this case there is no rotation because the function defines the direction relationship in all of the positions that would be encountered under a rotation, simultaneously. It is a smooth function. Edited May 17, 2016 by steveupson
Mordred Posted May 17, 2016 Posted May 17, 2016 (edited) Perhaps you better look at the math definition of a scalar vs a vector quantity. Particularly if your going to post incorrect claims.. Scalar. (of a quantity) having only magnitude, not direction. Look, it's math, it's either correct or it's incorrect, right? The model presents a description of a function that defines the relationship between a reference and some other direction. Call it what you will. Precisely note the need for a reference. Needing a reference is not invariant Anyone can describe direction to a reference. Everyone does so... No one describes a direction without a reference. Edited May 17, 2016 by Mordred
steveupson Posted May 17, 2016 Author Posted May 17, 2016 Perhaps you better look at the math definition of a scalar vs a vector quantity. Particularly if your going to post incorrect claims.. Scalar. (of a quantity) having only magnitude, not direction. Precisely note the need for a reference. Needing a reference is not invariant Anyone can describe direction to a reference. Everyone does so... No one describes a direction without a reference. This shouldn't be this painful for the two of us. Maybe we should give it a rest for a while and let some things sink in. We seem to keep repeating ourselves. Thanks for helping with this. I think we've made progress, but I'm sure that you don't really agree. cheers, steve
Mordred Posted May 17, 2016 Posted May 17, 2016 (edited) In all honesty the closest you will get is to study how to represent vectors in tensor form. However the tensor translations will have different coordinate transformation rules. The links I provided on tensors are some examples. You can easily Google more on vector calculus and by googling vector to tensor transformations. Type pdf at the end (it will give a better ratio of well written articles) These will guide you into mathematically describing a vector to perform a rotation or translational symmetry transformation. You can do a simple example yourself. Draw a line on a graph at a diagonal. Then take each point on the line and count three units right and 3 units up. When you connect those points you will have a vector with is identical to the original just translated to a new location. Mathematically describing this is covered in the links provided. Edited May 17, 2016 by Mordred
steveupson Posted May 17, 2016 Author Posted May 17, 2016 (edited) In all honesty the closest you will get is to study how to represent vectors in tensor form. However the tensor translations will have different coordinate transformation rules. The links I provided on tensors are some examples. You can easily Google more on vector calculus and by googling vector to tensor transformations. Type pdf at the end (it will give a better ratio of well written articles) These will guide you into mathematically describing a vector to perform a rotation or translational symmetry transformation. You can do a simple example yourself. Draw a line on a graph at a diagonal. Then take each point on the line and count three units right and 3 units up. When you connect those points you will have a vector with is identical to the original just translated to a new location. Mathematically describing this is covered in the links provided. Unfortunately, I don't really need any help with translation on a graph. Too bad for me, I guess, because you're so willing to help and all. What I need is some help with the actual questions that I asked. on edit> I'm being really unfair about this, and I apologize. I understand that my questions are not being answered because no one here knows the answers. That's not anyone's fault at all. But there's no reason to patronize me over it, either. Not at all I was quite serious. studiot, I've made an effort to look this over. You know what, though, I have to ask a very huge favor from you. Can you try and tell me exactly what you think I am claiming and then I'll try and clarify all of the misunderstandings. Everyone seems to be hearing me say things that are much different than what I've actually been saying. Can you please, in as much detail as you can, explain the model to me? Edited May 17, 2016 by steveupson
Mordred Posted May 17, 2016 Posted May 17, 2016 (edited) If you work it out I have been. You wanted to understand how to mathematically describe a vector as a tensor. So in order to explain we needed to determine what units will remain unchanged regardless of translational or rotational change. That value is the vector length. Which is the magnitude. Yes it's easy to describe a vector using Cartesian coordinates, Take an x,y graph. Using coordinate 0,0 as a start point then describe the opposite coordinates. You can extend this by adding the z component and for 4d use w. But all that is reference based. For example a vector with origin point 0,0,0 can be described by [latex] a=ax I+ay j+az k[/latex] [latex]a=\begin{pmatrix}a_x\\a_y\\a_z\end{pmatrix}[/latex] Now I just described in matrix form a vector. Did that somehow help if you don't understand how matrix and tensors work with coordinates i,j,k '(unit vectors)or in Einstien summations? Yes I described a vector, I can apply translation or rotations to this but the rules will vary according to those described by tensors. For example a 2d rotation of a line with 0,0 as the starting point will look like this in matrix form [latex]\begin{pmatrix}\acute{x}\\\acute{y}\end{pmatrix}\begin{pmatrix}\cos\theta&-sine\theta\\sine\theta&cos\theta\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}[/latex] The last set is your rotation Edited May 17, 2016 by Mordred
steveupson Posted May 17, 2016 Author Posted May 17, 2016 If you work it out I have been. You wanted to understand how to mathematically describe a vector as a tensor. So in order to explain we needed to determine what units will remain unchanged regardless of translational or rotational change. That isn't even close to any of the questions that I asked. There was a point at which I asked you to show me what you think I'm talking about, using the model, and you refused. I was hoping that by watching you work it through I could stop you at the point where we are not communicating and then try and correct that problem. I do appreciate that you want to help, and without a doubt this discussion has been fruitful to some degree, but unless we listen to each other we won't be able to solve this, imho. The questions that I have are basically three, and they are math questions. First, is there some way to algebraically express that the thing we are searching for is a function of a function? Next, is it possible to derive the function? Finally, if it isn't practical to define the function then is it possible to graph it?
Mordred Posted May 17, 2016 Posted May 17, 2016 (edited) You had better reword this last post in relation to the OP. In particular the thread title. I don't like wasting my time on confusion. The majority of your thread has been describing direction. In one example you specifically asked for tensors and vector relations. The physical property of direction does not appear on any list of physical properties. The reason for it not appearing on such lists seems to be unknown at this time, or least no accepted scientific source provides this reason. Without speculating as to what these reasons might be, there does seem to be a practical method that can be used in order to fill in this area of physics. There is a new model that illustrates the existence of a smooth function that establishes the relationship between two directions (orientated 45 degrees to one another) in three dimensions, simultaneously. It involves concurrent quantification of direction, without a metric, which can be understood as simultaneous finite rotations which commute in much the same manner as infinitesimal rotations commute in conventional plane geometry. This differs (mathematically and conceptually) from the usual (two dimensional) discrete finite rotations or infinitesimal rotations which are used to manipulate objects in Euclidean three space. In the new model, the relationship between the two directions is that they are 45 degrees to one another, but a similar model can be constructed for defining the relationship between any two directions. These are simply observations, not speculation. The new function is here: http://www.scienceforums.net/topic/95113-defining-a-new-function/ You want to describe direction, as a function. I already explained as well as others that direction isn't invariant. Vectors have both magnitude and direction. In this there is plenty of examples. Yet you choose to ignore why the scalar and not the direction quantity is invariant. The math I'm posting shows why this is the case. In the last post I made you can apply your rotations for 45 degrees in precisely the same manner as the link you provided. Yet you obviously don't want those details. Essentially what I've done above is shown A) how to describe a vector and b) the rotation rules in 2d. You can Google the 3d and 4d rotation rules. Edited May 17, 2016 by Mordred
steveupson Posted May 17, 2016 Author Posted May 17, 2016 You had better reword this last post in relation to the OP. In particular the thread title. I don't like wasting my time on confusion. The majority of your thread has been describing direction. In one example you specifically asked for tensors and vector relations. You want to describe direction, as a function. I already explained as well as others that direction isn't invariant. Vectors have both magnitude and direction. In this there is plenty of examples. Yet you choose to ignore why the scalar and not the direction quantity is invariant. The math I'm posting shows why this is the case I'll reword the post to say whatever you want it to say if we can answer at least the first of my three questions. You seem to think that we have some sort of disagreement or something over vectors or tensors or something. I assure that is not the case, and if I said anything at all to give that impression then I completely misspoke. Yes, it is worthwhile for me to learn much more about how these things relate to each other, but that is a side issue. I know that the stuff works. We went to the moon using these same techniques. I get that. I am not arguing with you about it. What this discussion is about is direction, with a specific focus on a method for quantifying it. Let me ask (politely) whether or not it is possible for us to use the proper notation to show what I am looking for? Math is very precise. I want that precision. I just don't know how to do it in a manner that anyone else would understand. What I'm after is something along the lines of: E is a function of alpha, and for every different direction the function changes.
Mordred Posted May 17, 2016 Posted May 17, 2016 (edited) Again you still have the problem. An arrow is easily described by the vector equation above. So you have direction. The only difference is you also have a magnitude. Direction is always relative to a reference. One can easily state go 20 degrees from point P. But then everyone has to agree that 0 degrees follows the positive y axis... You can have as many coordinates as you want but you still need a reference point. The above math uses 0,0,0 as the reference. This is the point you need to specify. You cannot describe direction without a reference. Angles from a starting coordinate is easily quantifying a direction but you have a coordinate system. The short answer is you need a reference to quantify a direction. It doesn't work to say go left if you don't know which way your facing. For example going upstream is a direction one that is against the current flow. Edited May 17, 2016 by Mordred 1
steveupson Posted May 17, 2016 Author Posted May 17, 2016 Again you still have the problem. An arrow is easily described by the vector equation above. So you have direction. The only difference is you also have a magnitude. Direction is always relative to a reference. One can easily state go 20 degrees from point P. But then everyone has to agree that 0 degrees follows the positive y axis... You can have as many coordinates as you want but you still need a reference point. The above math uses 0,0,0 as the reference. This is the point you need to specify. You cannot describe direction without a reference. Angles from a starting coordinate is easily quantifying a direction but you have a coordinate system. And what did i say that led you to believe that I have a problem with any of that? I'm curious, because I didn't intentionally say anything that was even remotely critical of any of that. What I am asking is whether or not it is possible to put my words into mathematical symbols. There is rigor that is required in order to do that. I think that no one is hearing what I am actually saying because they won't help me make the effort to express it algebraically. If the result is drivel then y'all have my permission to mock me mercilessly until I get to 1000 posts. Once we have the expression for you to evaluate, then we can go over all of the reasons why it won't work. But we can't (legitimately) evaluate the the expression until we know precisely what it is, can we? Of course not. What do you think the function is that I'm going on and on about? Can you tell me what you think I'm claiming?
Strange Posted May 17, 2016 Posted May 17, 2016 What this discussion is about is direction, with a specific focus on a method for quantifying it. Let me ask (politely) whether or not it is possible for us to use the proper notation to show what I am looking for? Math is very precise. I want that precision. I just don't know how to do it in a manner that anyone else would understand. What I'm after is something along the lines of: E is a function of alpha, and for every different direction the function changes. That is what vectors are for. These can use cartesian coordinates (i.e. N orthogonal axes, where N=3 for 3D space) or spherical coordinates or any other basis you choose (6 [or do I mean 8?] axes at 45 degrees, for example). There are simple functions to convert between each of these different coordinate systems, which is what "something along the lines of: E is a function of alpha, and for every different direction the function changes" sounds like. And if you are saying that the function changes, then that sounds a bit like you are talking about curried functions. There are, of course, many other ways of representing direction. For example, in 3D graphics we use the plane equation, ax+by+cz+d=0, to represent a surface because this makes it easy to calculate intersections, hidden surfaces, etc. This can be trivially converted into a direction by calculating the surface normal. Which is a vector: [a,b,c]. I don't mean to be impolite but ... as you don't have the mathematical knowledge to recognise that vectors (and operations on them) are the answer to your question, how would you recognise this Special Function you are looking for (even if someone were able to divine what it is from your vague descriptions)?
steveupson Posted May 17, 2016 Author Posted May 17, 2016 (edited) I don't mean to be impolite but ... as you don't have the mathematical knowledge to recognise that vectors (and operations on them) are the answer to your question, how would you recognise this Special Function you are looking for (even if someone were able to divine what it is from your vague descriptions) <see edit at bottom of post> I'm sorry, I do understand what you're saying, and I do understand how vectors, tensors, and coordinate systems work. I don't have the level of sophistication that others may have, but I do have a very solid understanding of the basics. I know when dot products and cross products should be used (not that I could do a dot product from memory). I know a little bit about polar coordinate systems and I'm pretty good with spherical trig (or at least I used to be). My problem is that I only know the things that I've used because that's the only things that I've bothered to learn. I have no formal training. This isn't about that. What we're doing doesn't change any of that stuff at all. I'm comfortable in knowing that those things can be always be used in the conventional manner, only we won't be using any of them here because we are not doing any of those operations. What we are trying to accomplish in this discussion is something that is in addition to those tools. What I need, what I cannot do without help, is compose a function. I just don't know how to do it. Your guess as to what I'm trying to do is only half right, and it also isn't precise enough. If we can get the basic form correct, then we shouldn't have any problem polishing it up since we'll both be looking at the same thing. We are not going to be using any coordinate systems. We'll have to figure out how to plug this into a frame later (or we may not need to). What we have are two angles. Angle alpha will always range between 90 to 0. Angle E will have a range that will vary according the magnitude of the direction. (Yes, I know that it's annoying for me to say it that way, I mean the size of the difference between the reference and the direction. If you have a preferred way to express that, please speak up and we'll use that term instead of magnitude.) When we are at 45 degrees, E will range between 0 to 90. When we are at 30 degrees, E will range between 0 to 60. It will always be twice the magnitude of the direction. The primary function is that for every angle E there will be a unique angle alpha. Let me stop here and see if we see eye to eye so far. On edit> I've taken a closer look at the circulation of vector fields that studiot wanted me to look at and I think I finally see what I have been missing. They do have a similar structure to what I was talking about, with one major difference. For almost a year I have been looking at this and assuming that there was a boundary there that would orient the thing in the frame and now it's finally dawning on me that there isn't one, it just continues to wrap around, so to speak. This is a problem. I fear it's a fatal one. I think that at this point I should retire from this thread unless there are any further developments or questions, which I don't expect that there will be. I don't know quite what to say to everyone. Will a thank you suffice? Edited May 17, 2016 by steveupson
Strange Posted May 17, 2016 Posted May 17, 2016 We are not going to be using any coordinate systems. We'll have to figure out how to plug this into a frame later (or we may not need to). What we have are two angles. Angle alpha will always range between 90 to 0. Angle E will have a range that will vary according the magnitude of the direction By having an angle, you have a coordinate system. Otherwise, what is your angle relative to? The primary function is that for every angle E there will be a unique angle alpha. If alpha is a unique function of E then you only have one variable (E) and so you can only express directions in two dimensions.
ajb Posted May 17, 2016 Posted May 17, 2016 By having an angle, you have a coordinate system. Otherwise, what is your angle relative to? I too am thinking like this. I am not sure what one could mean about a function of an angle without picking coordinates. 1
Mordred Posted May 17, 2016 Posted May 17, 2016 I can't even imagine describing a point in space without using some form of coordinate. At least not in any mathematics.
ajb Posted May 17, 2016 Posted May 17, 2016 I can't even imagine describing a point in space without using some form of coordinate. At least not in any mathematics. It depends what you mean by describing... the points 'exist' irrespective and independently of any chosen local coordinate system. But when you want to calculate something explicitly you almost always need to use local coordinates. Anyway, you can think of a point on a manifold as a map from the empty set to that manifold: no coordinates needed 1
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