Galilean:x' with respect to S'? AND SPECIAL BONUS x' = x-d?

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Re: Galilean:x' with respect to S'?

Postby Schrollini » Sat Jul 13, 2013 9:27 pm UTC

It looks like everyone's having a fun Saturday! I didn't read all the posts carefully, but I don't think I missed anything important. But if I did, please point it out.

Steve, it looks you still have a bunch of questions about manifolds. In general, they all seem to try to give the manifold too much structure. Just remember, the manifold only has points. Everything else, like coordinate systems, is additional structure that uses a manifold as a base.

Really, the best way to get a good understanding of these things is to do some examples. This is why I posted them! So if you haven't done it already, please go through the exercises and check that you understand the answers I've posted. If you understand these answers, you understand enough for us to continue. The next subject will be coordinate transformations, which I think will help further explain these concepts, so you don't have to have a perfect understanding yet.

So, please let me know if you understand the exercises, or ask questions if you don't.
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Re: Galilean:x' with respect to S'?

Postby steve waterman » Sat Jul 13, 2013 9:45 pm UTC

Schrollini wrote:It looks like everyone's having a fun Saturday! I didn't read all the posts carefully, but I don't think I missed anything important. But if I did, please point it out.


Yes, I hope you missed this. We probably posted at the same time-ish. My previous post to this.

I would like to hear your feedback/thoughts from things mentioned in it, please.

steve wrote:me - equality means having equal vector distances from their own (0,0,0) as the one another system.
as would be S'(2,0,0) = S(2,0,0) hence being a coordinate means being a vector from its own (0,0,0).

Galilean - equality means shared location
as would be S'(-1,0,0,) = S(2,0,0) - vt hence being a coordinate means being a point to you, NEVER A DISTANCE.

S(2,0,0) is a point in the Galilean mind-set and whereas for me,
S(2,0,0) is that vector of length of 2, starting from S(0,0,0) and going to S(2,0,0).

....................................................................................................................

I had added this to one of my posts to not double post and apparently after you had already read the
initial post.

added - Schrollini
I believe I should be able to follow your notation. If not, I will you ask you to please clarity. I will
allow your terms and their definitions too. Thanks for offering to walk us through your derivation.
If you wish diagrams, that will also be logical. I will, for the time being, suspend all requests for
answers to whatever is currently unanswered at this point in time, from you.

Meanwhile, I will respond to short posts. Longer posts, I will read and then put on the back-burner,
most likely, and unfortunately, not have the capacity/strength and energy to get to them, to eventually,
write back.
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"Be careful of what you believe, you are likely to make it the truth."
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Re: Galilean:x' with respect to S'?

Postby Schrollini » Sun Jul 14, 2013 4:36 am UTC

steve waterman wrote:I would like to hear your feedback/thoughts from things mentioned in it, please.

steve wrote:me - equality means having equal vector distances from their own (0,0,0) as the one another system.
as would be S'(2,0,0) = S(2,0,0) hence being a coordinate means being a vector from its own (0,0,0).

Galilean - equality means shared location
as would be S'(-1,0,0,) = S(2,0,0) - vt hence being a coordinate means being a point to you, NEVER A DISTANCE.

S(2,0,0) is a point in the Galilean mind-set and whereas for me,
S(2,0,0) is that vector of length of 2, starting from S(0,0,0) and going to S(2,0,0).

I have not introduced a vector space, so why are you talking about vectors? I have not introduced a metric, so why are you talking about distances? I have introduced only two concepts: a manifold and a coordinate system, and I'm still trying to understand if you understand them. Bringing up all sorts of other concepts isn't helpful.

Let me ask again, since you have yet to give me a straight answer: Are you satisfied with the solutions to the exercises that I posted? Everything else we do is contingent on these first steps, so we have to be on the same page. In fact, the result of one of the exercises will be used in the next step, so you really want to be okay with them if you want to be okay with the next step. I will not continue until you indicate that you've reviewed the exercises and are ready to proceed.
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Re: Galilean:x' with respect to S'?

Postby steve waterman » Sun Jul 14, 2013 12:24 pm UTC

steve wrote:me - equality means having equal vector distances from their own (0,0,0) as the one another system.
as would be S'(2,0,0) = S(2,0,0) hence being a coordinate means being a vector from its own (0,0,0).

Galilean - equality means shared location
as would be S'(-1,0,0,) = S(2,0,0) - vt hence being a coordinate means being a point to you, NEVER A DISTANCE.

S(2,0,0) is a point in the Galilean mind-set and whereas for me,
S(2,0,0) is that vector of length of 2, starting from S(0,0,0) and going to S(2,0,0).


Schrollini wrote:I have not introduced a vector space, so why are you talking about vectors? I have not introduced a metric, so why are you talking about distances?

I am came to realize this morning. re x' = x-vt.

x is a number whereas (x,0,0) is a point.

the equation that the Galilean logically supports is about POINTS
(x',0,0) = (x-vt,0,0)
I agree 100 percent with this equation

the equation that the Galilean states, is logically about DISTANCES/vectors
x' = x-vt
I disagree 100 percent with this equation

Schrollini wrote:Let me ask again, since you have yet to give me a straight answer: Are you satisfied with the solutions to the exercises that I posted? Everything else we do is contingent on these first steps, so we have to be on the same page. In fact, the result of one of the exercises will be used in the next step, so you really want to be okay with them if you want to be okay with the next step. I will not continue until you indicate that you've reviewed the exercises and are ready to proceed.

Yes, I am satisfied with the solutions. Yes. I have reviewed the exercises am I am ready to proceed. Please post your derivation. Yes, I accept your terms, definitions, understanding, diagrams, exercises. Please post your derivation
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Re: Galilean:x' with respect to S'?

Postby Schrollini » Sun Jul 14, 2013 8:27 pm UTC

Great. I'll post answers to exercises inline as spoilers, but I still recommend that you work through them yourself before checking them.

Coordinate Transformations

We can consider multiple coordinate systems for a given manifold. For concreteness, let's consider two, which we'll call f and g. And for even more concreteness, let's look at the two I've sketched here:
translation1.png

Recall that this image represents the manifold and the blue and red bits are the images of certain coordinates.

Exercise 5: Find f-1(P) and g-1(P).
Spoiler:
f-1(P) = (2,1) and g-1(P) = (0.5, 1.5). Thus, we say that (2,1) are the coordinates of P in coordinate system f and (0.5, 1.5) are its coordinates in g.


Moving back to the abstract: For real numbers x, y, u, and v, f(x,y) and g(u,v) are both points in manifold. Recall that manifolds have very little structure, so basically all we can say about these two points is whether they are the same or not. If they are the same, this suggests a relationship between the coordinates. This is formalized in:

Definition: A coordinate transformation from coordinate system f to coordinate system g is a mapping h from ℝ2 to ℝ2 such that g(h(x,y)) = f(x,y).

This is rather dense, so let me restate it in other words. A coordinate transformation is a function that takes the coordinates of a point in one coordinate system to the coordinates in another system. So if f(x,y) = g(u,v), the coordinate transformation h will have h(x,y) = (u,v).

We can write h explicitly in terms of f and g. Start with f(x,y) = g(u,v) and take g-1 of both sides. From exercise 1, we know that that this gives us g-1(f(x,y)) = (u,v). Thus, we see that the coordinate transformation from f to g is just the inverse of g composed with f.

Since a coordinate transformation is a mapping from real numbers to real numbers, we can explicitly write it down in arithmetic. For the case posted above, the transformation is g-1(f(x,y)) = (x-1.5, y+0.5).

Exercise 6: Verify the results of the previous exercise using this formula.
Spoiler:
2 - 1.5 = 0.5 and 1 + 0.5 = 1.5. (These exercises aren't particularly hard.)


This type of transformation is called a translation. This is one example of a convention that's a bit misleading -- coordinate transformations are given names that suggest actions. Don't let this fool you. I didn't "translate" a coordinate system or the manifold. "Translation" is just a name. Anyway, a translation is a coordinate transformation in which both coordinates are offset by a constant amount. In general, a translation can be written as h(x,y) = (x+a, y+b) for some real numbers a and b.

Exercise 7: Write the coordinate transformation h between two coordinate systems separated by a translation, if the point at the origin of the first coordinate system has coordinates (5,0) in the second.
Spoiler:
The condition stated in the problem is that h(0,0) = (5,0). Plugging this into the general form for a translation, we see a = 5 and b = 0, so the transformation is h(x,y) = (x+5, y).


This is all for now. The good news is that these are all the math definitions we will need. We still have a way to go, but it's all just working out the consequences. Let me know if you understand the definitions and the exercises. Once we're okay, we'll move on to the next step, in which I'll do my best to rigorously define the sloppy notation that we tend to use in practice.

Edit: Fix markup. (Thanks, Jose.)
Last edited by Schrollini on Mon Jul 15, 2013 2:43 am UTC, edited 1 time in total.
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Re: Galilean:x' with respect to S'?

Postby steve waterman » Sun Jul 14, 2013 10:16 pm UTC

Anyone have some feedback regarding these suggested equations? - ivnja?
(x',0,0,) = (x-vt,0,0)
(x+vt,0,0,) = (x',0,0)


ADDED -
S'(0,0,0) to S'(x',0,0) = S(0,0,0) to S(x,0,0)
x' = x

now finished any edits/adding

ooops...I need to add again, please excuse.
M(x',0,0,) = S(x-vt,0,0)
M(x+vt,0,0,) = S'(x',0,0)


where M would mean onto top of manifold M - the "Mappee" system
and S or S' is the appropriate "Sender" system
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Re: Galilean:x' with respect to S'?

Postby steve waterman » Sun Jul 14, 2013 11:39 pm UTC

Schrollini

I appreciate the excellent job you are doing. You are giving comprehensive definitions and leaving a great paper trail for easy reference. I even like your spoiler approach too. Please keep going. So, posting it in sections is fine. Likely, I will hold any counter-point crap/objections/etc until after I see the whole thing. I will read that which you post, and most likely want to hold any comments until the whole derivation is completed and posted. So, excellent job on the style and the set up. When parts of your posted derivation perplex me, I will first be able to use what you have posted as reference for any math pertinent to that notation.

Which are you going to be deriving...?
1 the point on top of manifold M at (x',0,0) = S(x-vt,0,0)
2 the distance from S'(0,0,0) to S'(x',0,0) = the distance from S(0,0,0) to S(x-vt,0,0)
3 x' = x-vt
4 x' = f(x-vt)
5 other
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Re: Galilean:x' with respect to S'?

Postby ucim » Sun Jul 14, 2013 11:53 pm UTC

Shrollini, There may be a typo (misplaced italics command) in your answer for exercise 7. (edit - fixed)

Great job so far, btw. :)

Jose
Last edited by ucim on Mon Jul 15, 2013 6:23 pm UTC, edited 1 time in total.
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Re: Galilean:x' with respect to S'?

Postby oauitam » Mon Jul 15, 2013 2:22 pm UTC

steve waterman wrote:Likely, I will hold any counter-point crap/objections/etc until after I see the whole thing. I will read that which you post, and most likely want to hold any comments until the whole derivation is completed and posted.

NO.

This is exactly the wrong thing to do.
He is posting step by step for a specific reason, not cos it's just happening to come out that way.

Do not hold any questions, counter-point crap, queries, objections, clarifications, i-dont-care-what until you see the whole thing. Post them immediately.

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Re: Galilean:x' with respect to S'?

Postby Schrollini » Mon Jul 15, 2013 3:17 pm UTC

oauitam wrote:Do not hold any questions, counter-point crap, queries, objections, clarifications, i-dont-care-what until you see the whole thing. Post them immediately.

I have asked Steve to hold objections to my definitions until the end, and I appreciate him doing so. I know this is hard for him - he's worked with a different set of definitions, but he's been willing to set those aside and work with mine. I really appreciate that.

Also, I've asked him not to bring up questions that go beyond what I've defined. It's a natural question to ask if thing A can do task X, but I'm trying not to get sidetracked. If task X is important, I'll introduce it at some point. If not, we can discuss it at the end.

That said, it would really be better, as oauitam says, to address questions about the definitions and exercises now, rather than at the end. The Galilean transformation is a coordinate transformation, and it'll be very frustrating to discover that a question about it is actually a general question about coordinate transformations, but is actually a question about coordinate systems. Now's the time to work out the fundamental questions so that we can later use the fundamentals without confusions.

Not sure if your question is fundamental or irrelevant? The best way to see is to do the exercises. If you can do them, and you get the same answer I give, you should understand enough to continue, even if you don't understand everything. If you don't understand what's being asked, or if you get a different answer, then there is a fundamental confusion that I'd much rather address now.

I'm going to start writing the next lesson, which will feature some more example coordinate transformations. This will be a very good place to check your understanding. But don't be afraid to ask questions before that post goes up. I can address those first (or tell you to wait on the coming lesson, if I'm planning to get there).
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Re: Galilean:x' with respect to S'?

Postby steve waterman » Mon Jul 15, 2013 3:28 pm UTC

Schrollini - Which EQUATION are you going to be deriving...PLEASE?
1 the point on top of manifold M at (x',0,0) = S(x-vt,0,0)
2 the distance from S'(0,0,0) to S'(x',0,0) = the distance from S(0,0,0) to S(x-vt,0,0)
3 x' = x-vt
4 x' = f(x-vt)
5 other

added -
You really need to answer this now, or MY participation in YOUR derivation of "what equation"? is over.
Days have passed and I asked a half dozen times to simply post the derivation. Hopefully,
this thread will be allowed to stay open for some others to share their own opinions, given my refusal
to continue this, under these conditions. I think it is a fair question..."what is your derivations out to prove?"

added-
I observe this too. In your one diagram, of your recently related derivation posts.
point (x',0,0) = point (x-vt,0,0)
distance x' = distance x
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Re: Galilean:x' with respect to S'?

Postby beojan » Mon Jul 15, 2013 4:27 pm UTC

Steve, what you have been doing is a little like someone coming to these fora and claiming that 5 + 7 = 10, claiming the meaning of the symbols has been misinterpreted when people disagree.

The only way forward is to teach you what addition is, or in this case, teach you what coordinate systems, and points, and coordinate transforms are, and make sure you understand them. In short, what Schrollini is doing is not (really / only) a derivation of the Galilean transform, but rather a course in coordinate geometry, that will (hopefully) eventually lead to you understanding the Galilean transform, and coordinate transformations in general. However, this does mean you need to follow Schrollini's course, and do so in good faith, without attempting to sidetrack the discussion or preempt the course.

Note: Preempting the course is when you ask repeatedly what equation he is going to derive, before much of what makes said equation(s) meaningful has even been introduced.

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Re: Galilean:x' with respect to S'?

Postby steve waterman » Mon Jul 15, 2013 4:46 pm UTC

POINTS
(x',y',z') = (x,y,z) +/- (a,b,c)

DISTANCES
x = x'
y = y'
z = z'
t = t'

therefore, expressed as distances,
x' ≠ x-vt
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Re: Galilean:x' with respect to S'?

Postby ivnja » Mon Jul 15, 2013 5:05 pm UTC

Steve, please be patient. Like you said, 25 years you've been working on this. Don't let a couple days of slower than you'd like posting ruin this session. Schrollini's a busy man; give him a chance to respond. He said in his first post that he was going to derive the Galilean transform, so he's going to get to x' = x - vt, but it's going to take a few steps.
Hi you.
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Re: Galilean:x' with respect to S'?

Postby steve waterman » Mon Jul 15, 2013 5:38 pm UTC

ivnja wrote:Steve, please be patient. Like you said, 25 years you've been working on this. Don't let a couple days of slower than you'd like posting ruin this session. Schrollini's a busy man; give him a chance to respond. He said in his first post that he was going to derive the Galilean transform, so he's going to get to x' = x - vt, but it's going to take a few steps.


Schrollini can only come to (x',0,0) = (x-vt,0,0) as he is dealing with a point mindset,
however, with a distance mindset, x' = x. x, being the abscissa.

ab·scis·sa
[ab-sis-uh]
noun, plural ab·scis·sas, ab·scis·sae [ab-sis-ee] . Mathematics.
(in plane Cartesian coordinates) the x-coordinate of a point: its distance from the y-axis measured parallel to the x-axis.
http://dictionary.reference.com/browse/abscissa
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Re: Galilean:x' with respect to S'?

Postby Schrollini » Mon Jul 15, 2013 7:03 pm UTC

Steve, if you're so desperate to see the Galilean transformation, here it is, in the notation we've introduced so far:

The Galilean transformation is the coordinate transformation h parameterized by real number v given by hv(x,y) = (x-vy, y).

It's just a definition, so there's nothing to prove, or for that matter to even argue about. But to understand it, you must understand what a coordinate transformation is, and to understand that, you must understand what a coordinate system is, and to understand that, you must understand what a manifold is. This is why I started where I did. Furthermore, to understand why physicists think it is the bee's knees (to say nothing of the cat's pajamas), you need to understand what spacetime and reference frames are. I'm getting to them, honest.

But first:

A Note on Notation

The notation I've been using for coordinate systems and transformations is a bit heavy; that is, there are a lot of symbols flying around. Because coordinate systems are used as the basis for so much other math (and physics), we tend to use a somewhat shorter, if sloppier, notation. Instead of referring to a coordinate system by a name, like f or g, we give the coordinates themselves names, and refer to the coordinate system by the coordinates themselves. So instead of talking about f or g, I'd talk about the xy system or the uv system. Instead of referring to a point in the manifold as f(x,y), we'd refer to it as (x,y). Different notation, but the same meaning.

Admittedly, this is a bit sloppy. If I were to write (2,5), it's not clear a priori whether I mean the ordered pair (2,5), the point f(2,5), or the point g(2,5). In practice, it's generally perfectly clear from context, but you must always check to make sure you're choosing the right meaning. To try to keep things clear, I'll use the notation (x=2, y=5) or (u=2, v=5) for these latter two cases.

If we use this notation in our favorite example, it might look more familiar:
translation2.png
translation2.png (9.75 KiB) Viewed 4419 times

With our new notation, we'd write (x=2, y=1) = P = (u=0.5, v=1.5).

Since we're not giving the coordinate systems explicit names, we can't write the coordinate transformation from one to the other as g-1(f). Instead, we'll just give the coordinate transformation as a set of equations relating the coordinates. For this example, we'd write u = x-1.5, v = y+0.5. These are just a couple of algebraic equations, but you must remember that, behind the scenes, there are two coordinate systems and a manifold! You can do the math without understanding this, but you won't understand what it means.

Exercise 8: Write a general translation between two coordinate system in the new notation.
Spoiler:
Let the two systems be xy and uv. The general translation between them is given by u = x-a, v = y+b, for real numbers a and b.


Several Examples, Chosen with Malice Aforethought

The following is an example of a rotation:
rotation.png

This specific transformation is given by u = (x+y)/√2, v = (-x+y)/√2. In general, the rotation is parameterized by a single real number θ, and is given by u = x cos θ + y sin θ, v = -x sin θ + y cos θ. The specific example shown has θ = π/4.

Exercise 9: Find the coordinates of P = (x=2, y=1) in the uv system in both the general and specific cases.
Spoiler:
u = 2 cos θ + sin θ = 3/√2, v = -2 sin θ + cos θ = -1/√2


Here's an example of a single-axis skew transformation:
skew.png
skew.png (10.87 KiB) Viewed 4419 times

This specific transformation is given by u = x - y/2, v = y. In general, the skew is parameterized by a single real number m, and is given by u = x - m y, v = y.

Exercise 10: Find the equation for the v axis in the xy coordinate system.
Spoiler:
The v axis has u = 0. Putting this into the coordinate transformation, we get x = m y. In the specific case, m = 1/2, so the equation becomes y = 2x.


Finally, this is a shear transformation:
shear.png

A shear transformation has a single parameter η: u = x cosh η - y sinh η, v = -x sinh η + y cosh η. sinh and cosh are the hyperbolic sine and cosine functions, respectively.

Exercise 11: Given that P = (x=2, y=1) lies on the u axis, find η.
Spoiler:
The u axis is given by v = 0 = 2 cosh η - sinh η. Therefore, we have tanh η = 2 and η = 0.549.


That's it for now. I'm traveling overnight, so you won't hear from me until sometime Tuesday. Next time, we'll look at how physicists use coordinate systems and coordinate transformations. So this would be a good time to review the exercises and make sure you have the math part down pat. There are plenty of capable people around to answer questions on the exercises until I get back.

Note to mod: For some reason, two of these images show up with scrollbars, which is rather annoying. If there's something you can do to get rid of them, please do it!
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Re: Galilean:x' with respect to S'?

Postby WibblyWobbly » Mon Jul 15, 2013 7:36 pm UTC

Schrollini wrote:Note to mod: For some reason, two of these images show up with scrollbars, which is rather annoying. If there's something you can do to get rid of them, please do it!


If I click on the image that has a scroll bar, it loads the whole image.

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Re: Galilean:x' with respect to S'?

Postby steve waterman » Mon Jul 15, 2013 7:51 pm UTC

note to ST - Please allow others to comment re x being a distance and (x,0,0) being a point.
Why must his derivation be exclusive content of this thread?
I appreciate that you protected me from a possible mountain of messages...too many to handle.

However, there was a huge change in awareness since the thread started, recently; stated above.

Waiting x more days to hear feedback from others based upon Schrollini's derivation exposure time schedule will be too frustrating to deal with, and I won't. To others, DO PLEASE WAIT, posting anything at all until I find out that that is going to be okay. Thanks to any currently holding their tongues.

friday at 1:24
Okay. I will observe your derivation with your definitions.
Please do not attempt to force ME to jump through hoops to see YOUR presentation.
Use your defined terms and walk us all through, please.
I will avoid answering your/all questions/posts until you post your stand-alone derivation, okay?
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Re: Galilean:x' with respect to S'?

Postby eSOANEM » Mon Jul 15, 2013 9:53 pm UTC

steve waterman wrote:note to ST - Please allow others to comment re x being a distance and (x,0,0) being a point.
Why must his derivation be exclusive content of this thread?
I appreciate that you protected me from a possible mountain of messages...too many to handle.


You answered your own question.

Schrollini's posts are the best way for you to learn this because he is taking the care to explain this properly and patiently. We are bowing to ST's wishes because we realise that Schrollini's explaining it better than we would.

I will however say one thing because it seems to be a fundamental misunderstanding of what it means to have two co-ordinate systems S and S' and I believe it can be addressed without skipping ahead of Schrollini's course.

steve waterman wrote:POINTS
(x',y',z') = (x,y,z) +/- (a,b,c)

DISTANCES
x = x'
y = y'
z = z'
t = t'

therefore, expressed as distances,
x' ≠ x-vt


x and x' (and also y & y' and z & z') are both distances. They are also (in a cartesian system), distances in the same direction. They are however not distances between the same two points. As such they do not have to be the same length in both co-ordinate systems.

x is the distance between (in schrollini's notation where S is a function of an ordered quadruple (because we're in 3+1D here) giving a point e.g. P=(x,y,z,t) (for specific values of x, y, z and t)) S(0,0,0,t) and S(x,0,0,t).

x' is the distance between S'(0,0,0,t) and S'(x',0,0,t).

Now, if we're doing a co-ordinate transform between S and S', we're interested in the case where S(x,y,z,t)=S'(x',y',z',t) (note, t is not primed because, in galilean relativity, time is absolute and the transformation for it is simply t=t' and so I can eliminate on co-ordinate straight away).

The thing is, this does not imply that S(0,0,0,t)=S'(0,0,0,t) and this is, generally not true. In fact, in its usual form, the galilean transformation assumes that this is only true at t=0 and that subsequently the distance between S(0,0,0,t) and S'(0,0,0,t) is vt in the x direction (and 0 in all other directions) i.e. that looking at constant-time slices, the origin of one co-ordinate system appears to be moving away from the other at the rate of v [units of distance]/[unit of time].

If there is some distance between S(0,0,0,t) and S'(0,0,0,t) it follows then (assuming again that S(x,y,z,t)=S'(x',y',z',t)) that the distances between S(0,0,0,t) & S(x,0,0,t) and between S'(0,0,0,t) & S'(x',0,0,t) will not be equal.

I will refrain from going further because that is for Schrollini to do. I believe I have explained this distinction as well as I can without skipping ahead or sacrificing too much rigour although I may be mistaken in this.
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Re: Galilean:x' with respect to S'?

Postby ucim » Mon Jul 15, 2013 11:08 pm UTC

Steve - I would recommend that you not jump in with distances just yet. Distance isn't meaningful without some agreement as to what it means (in math, it requires a metric). (First sentence in that link: In mathematics, a metric or distance function is a function that defines a distance between elements of a set.), and remember a manifold is a set (which happens to have points as elements).

There are commonly used metrics, and often people use a shorthand that balls up a metric with a manifold and a coordinate system, and talks about points as if they were something else. Take the pieces Schrollini is giving you, one at a time, and understand each of them separately first.

If you do it Schrollini's way, you will soon see what a metric is and how it is applied. You will then be able to use the concept without falling into the trap of treating it like something else that it isn't, because that's what you are used to.

Examples of metrics between places in a city:
Spoiler:
  • the pythagorean metric (as the crow flies)
  • the taxicab metric (diagonals not allowed - must stay on the roads)
  • the Sunday midnight time metric (the time it takes between places, which depends on road conditions (no traffic on Sundays at midnight))
  • the Santa metric (how far apart in time Santa visits one place than another)
    I can also invent some silly ones, but the point is that the choice of metric depends on what you want to do. You'll notice that some of these metrics depend on things other than position. That's ok. In fact, it's an important concept - separating metric from (geographical) position.
    Try coming up with some other metrics for places in a city.
Examples of metrics between positions of a Rubik's cube:
Spoiler:
  • minimum number of moves required to get from one position to another (requires, of course, agreement on what constitutes a "move")
  • weighted minimum number of moves required to get from one position to another (certain kinds of moves "cost more")
    Can you think of any others? There is a multi-dimensional one that comes to mind.
Metrics applied to my pile of books:
Spoiler:
  • The absolute value of the difference between the slot numbers of two books.
What is wrong with that example? How would you fix it?
(hint - I left something important out)
Having contemplated these things, I would recommend that you put them aside, and return to them when Schrollini introduces metrics. For now, the point is that you are letting your preconceptions mislead you astray. Let Schrollini show you the components in the order he has in mind, putting aside what you already know. If you do, everything he says will be very clear, and in a few steps, you will get to your favorite transform, and it, too, will be crystal clear.

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Re: Galilean:x' with respect to S'?

Postby steve waterman » Tue Jul 16, 2013 1:47 pm UTC

if someone here at xkcd grasped that x represents distance in the Galilean equation and not the point (x,0,0) then I will continue my posting to this thread.

Otherwise, I have presented my challenge, and there was no interest in the ramifications of my new insight.

This insight was a Eureka moment ( for me ) after all this time and exposure via xkcd. I feel I might as well have not said this, as it certainly seems that no one else has yet to (fully) grasp the/any significant to this understanding.

I now wonder if other people's posts are being screened first or if, most likely, no one has a positive comment on my stuff. I am rapidly running out of things to say on this thread.

overview
x'= x-vt refers to abscissa vector distances from their own origin and not merely to points at (x',0,0) and (x,0,0).
x' = x and (x',0,0) = (x-vt,0,0)


added -

Image

Image
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Re: Galilean:x' with respect to S'?

Postby PM 2Ring » Tue Jul 16, 2013 5:15 pm UTC

Could you clarify this for me, Steve?
steve waterman wrote:1 the point on top of manifold M at (x',0,0) = S(x-vt,0,0)

A manifold is a set of points, so in standard mathematical language we talk about points being in manifold M. So what do you mean by a point on top of manifold M?


steve waterman wrote:if someone here at xkcd grasped that x represents distance in the Galilean equation and not the point (x,0,0) then I will continue my posting to this thread.

Otherwise, I have presented my challenge, and there was no interest in the ramifications of my new insight.

This insight was a Eureka moment ( for me ) after all this time and exposure via xkcd. I feel I might as well have not said this, as it certainly seems that no one else has yet to (fully) grasp the/any significant to this understanding.


Maybe we're not commenting because we don't want to interfere with Schrollini's presentation (and we fear the wrath of ST :) ).

Yes, it's true that x is not the point (x,0,0), it's a coordinate of that point (in a given coordinate system), and so yes it does represent a distance. But please bear in mind that the fact that x is a coordinate is more fundamental than the notion of x being a distance. If we're given two sets of coordinates (in a given coordinate system) we can determine whether or not they refer to the same point, but if they refer to two different points we need some more information about the manifold and the coordinate system before we can determine the actual distance between those two points.

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Re: Galilean:x' with respect to S'?

Postby JudeMorrigan » Tue Jul 16, 2013 5:37 pm UTC

Steve, we all figured out pretty quickly in the epic thread was that the problem was that you didn't really understand what people were saying with their notation. For what it's worth, it seems like you're a lot closer having had your eureka moment, but you're really not entirely there yet. I promise you that there are good reasons for the notation being used in the fashion it's being used. I heartily recommend sticking around, patiently and honestly going through Schrollini's presentation with an open mind. If you do so, you may have another eureka moment or two when all is said and done. But in order to do so, you'll have to set aside your preconceived notions as to what people are doing with the gallilean and why they're doing it that way.

(edited to fix typo)
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Re: Galilean:x' with respect to S'?

Postby eSOANEM » Tue Jul 16, 2013 5:38 pm UTC

steve waterman wrote:if someone here at xkcd grasped that x represents distance in the Galilean equation and not the point (x,0,0) then I will continue my posting to this thread.

Otherwise, I have presented my challenge, and there was no interest in the ramifications of my new insight.

This insight was a Eureka moment ( for me ) after all this time and exposure via xkcd. I feel I might as well have not said this, as it certainly seems that no one else has yet to (fully) grasp the/any significant to this understanding.

I now wonder if other people's posts are being screened first or if, most likely, no one has a positive comment on my stuff. I am rapidly running out of things to say on this thread.

overview
x'= x-vt refers to abscissa vector distances from their own origin and not merely to points at (x',0,0) and (x,0,0).
x' = x and (x',0,0) = (x-vt,0,0)


added -

Image

Image


Steve, your diagrams show that you have misunderstood what the Galilean is trying to do. x and x' are the co-ordinates (in the x-direction) of the same point but from different origins. They are not equal co-ordinates of different points from different origins.

Also no, people's posts are not being screened. We are simply trying to interfere as little as possible.
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Re: Galilean:x' with respect to S'?

Postby SecondTalon » Tue Jul 16, 2013 5:49 pm UTC

steve waterman wrote:note to ST - Please allow others to comment re x being a distance and (x,0,0) being a point.

Problem - The only one who is consistently getting through your thick skull and making you see anything remotely resembling reason and using the terminology the same way as the rest of the planet is Schrollini. I don't know how, I don't know why, I don't really care why. When others get involved, it becomes three or four people saying the same thing in slightly different ways to the point where I even understand what they're talking about and.. you breeze right over it, at best picking up on a logic or mathematical error or something in one of the dozens of examples and explanations given by the three+ people, and using that to .. well, essentially declare everyone as being Educated Stupid and reverting back to your wackyland definitions that only you seem to understand.

Hence, my desire to keep this as just you and Schrollini until such time as either Schrollini grows frustrated or it appears that you are actually listening to what Schrollini is saying the first time and not arguing with him for six or seven posts about what exactly x means in this particular equation.


And no, no one's screening anything. Most people are simply respecting my redtexting and staying out of it.

Seriously, I know shit about math and when I read your stuff and think "that's not right" you're really out in left field. So far out you're playing soccer. And seeing as that's a baseball metaphor, playing soccer is a Very Bad ThingTM.
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Re: Galilean:x' with respect to S'?

Postby steve waterman » Tue Jul 16, 2013 8:41 pm UTC

SecondTalon wrote:
steve waterman wrote:note to ST - Please allow others to comment re x being a distance and (x,0,0) being a point.

Problem - The only one who is consistently getting through your thick skull and making you see anything remotely resembling reason and using the terminology the same way as the rest of the planet is Schrollini. I don't know how, I don't know why, I don't really care why. When others get involved, it becomes three or four people saying the same thing in slightly different ways to the point where I even understand what they're talking about and.. you breeze right over it, at best picking up on a logic or mathematical error or something in one of the dozens of examples and explanations given by the three+ people, and using that to .. well, essentially declare everyone as being Educated Stupid and reverting back to your wackyland definitions that only you seem to understand.

Hence, my desire to keep this as just you and Schrollini until such time as either Schrollini grows frustrated or it appears that you are actually listening to what Schrollini is saying the first time and not arguing with him for six or seven posts about what exactly x means in this particular equation.


And no, no one's screening anything. Most people are simply respecting my redtexting and staying out of it.

Seriously, I know shit about math and when I read your stuff and think "that's not right" you're really out in left field. So far out you're playing soccer. And seeing as that's a baseball metaphor, playing soccer is a Very Bad ThingTM.


ST wrote:reverting back to your wackyland definitions that only you seem to understand.


The only term I used, I also supplied a url with the definition for; abscissa, and that was very recently.

the abscissa of x = the abscissa of x'
(x',0,0) = (x-vt,0,0)

So...I also start with what the Galilean given, and I only have these two equations.
What term/concept/notation do you not completely understand in these two equations?
That is, you can totally ignore everything else on these threads up to now.
All you/anyone now gets to critique is JUST these two equations. So, there are no other terms ( mine or otherwise) , nor definitions nor statements nor other than these 2 diagrams, to object to...there are none.

These are either true or false. There is no subjectivity involved, only math. Someone show me why either is not 100 percent true...in their opinion. Or, if by some miracle, you think one the two equations is actually correct, you might post that opinion too.
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Re: Galilean:x' with respect to S'?

Postby SecondTalon » Tue Jul 16, 2013 8:59 pm UTC

No, not that. Basic stuff that you say, someone says "Wait, are you using that right because it should go blah blah blah" and then you say "Oh, no, I'm using that term to mean blah blah blah" and then 15+ people immediately replying "Dude, that's.. no, that word doesn't mean that. At all"

That's what I mean by "wackyland definitions". Like if I decided that everything that was put in an oven and heated was caked. That I spent the afternoon caking in the kitchen and made a couple dozen cookies and a roast. People might pick it up on context what it is I'm meaning, but I'm using a word wrong based on.. I don't know what. A misunderstanding of something I read years ago and never corrected, maybe? And then reacted with hostility to the idea that I should use the word "baked" instead as caked is perfectly fine and we all just need to agree that caked means put in an oven and heated and we can move on to discussing this Duck a l'Orange recipe I'm having problems with, particularly when I should be adding the cocoa powder.

...

In case you don't get the joke there, cocoa powder is not an ingredient used in Duck a l'Orange. At least, not usually. Might actually work, now that I think about it....

Also, I'm an asshole. Can't ever forget that part.

Anyway, that's my point. Schrollini is working with you, trying to get you to understand the definitions and functions used by the rest of the planet as once you have those down, we can move on to what your actual problem is with .. whatever.. as you'll all have a common language with which to discuss it.

Because right now you're just angrily yelling Vietnamese in the middle of Gulfport, Mississippi and wondering why no one's understanding you.
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Re: Galilean:x' with respect to S'?

Postby steve waterman » Tue Jul 16, 2013 9:36 pm UTC

SecondTalon wrote:No, not that. Basic stuff that you say, someone says "Wait, are you using that right because it should go blah blah blah" and then you say "Oh, no, I'm using that term to mean blah blah blah" and then 15+ people immediately replying "Dude, that's.. no, that word doesn't mean that. At all"

That's what I mean by "wackyland definitions". Like if I decided that everything that was put in an oven and heated was caked. That I spent the afternoon caking in the kitchen and made a couple dozen cookies and a roast. People might pick it up on context what it is I'm meaning, but I'm using a word wrong based on.. I don't know what. A misunderstanding of something I read years ago and never corrected, maybe? And then reacted with hostility to the idea that I should use the word "baked" instead as caked is perfectly fine and we all just need to agree that caked means put in an oven and heated and we can move on to discussing this Duck a l'Orange recipe I'm having problems with, particularly when I should be adding the cocoa powder.

...

In case you don't get the joke there, cocoa powder is not an ingredient used in Duck a l'Orange. At least, not usually. Might actually work, now that I think about it....

Also, I'm an asshole. Can't ever forget that part.

Anyway, that's my point. Schrollini is working with you, trying to get you to understand the definitions and functions used by the rest of the planet as once you have those down, we can move on to what your actual problem is with .. whatever.. as you'll all have a common language with which to discuss it.

Because right now you're just angrily yelling Vietnamese in the middle of Gulfport, Mississippi and wondering why no one's understanding you.


I was speaking Vietnamese/distance mindset before, in a desperate attempt to parse all the xkcd comments/point mindset. I now am only speaking two equations. Are they that complex, that no one even has a clue what the math means in either, so much so, that they cannot even formulate a question/drop of feedback? I am at a total loss regarding what could possibly be unclear wrt
the abscissa of x = the abscissa of x' [ distance mindset ]
(x',0,0) = (x-vt,0,0) [ point mindset ]
both equations are true
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Re: Galilean:x' with respect to S'?

Postby Schrollini » Wed Jul 17, 2013 1:48 am UTC

steve waterman wrote:I was speaking Vietnamese/distance mindset before, in a desperate attempt to parse all the xkcd comments/point mindset. I now am only speaking two equations. Are they that complex, that no one even has a clue what the math means in either, so much so, that they cannot even formulate a question/drop of feedback? I am at a total loss regarding what could possibly be unclear wrt
the abscissa of x = the abscissa of x' [ distance mindset ]
(x',0,0) = (x-vt,0,0) [ point mindset ]
both equations are true

I haven't commented because I've been busy changing hemispheres. (Hey, did you know it's winter down here?) But I have two thoughts about why others haven't said anything:

1) An equation without context is meaningless. I can't say whether E = mc2 is true or false without first defining that E is the energy, m the rest mass, and c the speed of light. You haven't defined what you mean by x and x', so the truth value is undetermined. (Furthermore, you know from my previous post that the notation (x,y,z) can be a bit ambiguous, so you should specify what exactly you mean when you write an ordered pair.) Now, there's a set of standard definitions and notations that most people use, so we often talk with each other we use these implicitly. You've shown pretty conclusively that you're using your own set of definitions and notations, so it should be no surprise that no one will comment on an equation you give without rigorously defined symbols.

This is the whole point of the exercise I'm trying to lead. If you understand and use the standard definitions and notation, the rest of us will be able to understand your point much more easily. But you seem to be resisting this quite valiantly.

2) Your point is trivial. If your point is that there's a difference between points and distances, well, we all knew that already. As several people have pointed out, points come from a manifold and distances from a metric. Don't expect lavish praise for coming to this conclusion.

Anyway, you don't seem to be showing much interest in my curriculum. I'll go ahead and post that last bit for completeness, but I'm not holding much hope that you'll do the work necessary to understand it. I'd love to be proven wrong, though.
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Re: Galilean:x' with respect to S'?

Postby ucim » Wed Jul 17, 2013 2:28 am UTC

steve waterman wrote:This insight was a Eureka moment ( for me ) after all this time and exposure via xkcd. I feel I might as well have not said this, as it certainly seems that no one else has yet to (fully) grasp the/any significant to this understanding.
I did not comment because I was trying to stay out of the way. However I will say this:

1: Yes, that was a good Eureka moment for you. There are however a few more Eureka moments lurking out there for you. Schrollini is trying to lead you there. Follow him and your time will be rewarded. Not instantly, because it takes time to figure out how to apply your Eureka moment.

2: The key insight that I hope your Eureka moment brought you is that points and distances are not the same, and are not even properties of the same kind of thing. In your mind, you need to separate (and isolate!) "Manifold" (collection of points), "Real number line" (ordered set of real numbers), and "mappings" (ways to get from one to the other). Notice that "distance" isn't in here at all. It is yet another separate concept.

3: You need all the upcoming Eureka moments in order to see what we mean when we talk about the Galilean (or any other) transformation. Schrollini will take you there if you let him. Then you can put them together again, and you'll see the source of the confusion. But you can't put them together until you've taken them apart!

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Re: Galilean:x' with respect to S'?

Postby Schrollini » Wed Jul 17, 2013 4:24 am UTC

Spacetime and Reference Frames

Thus far, we've been traipsing through the abstract world of pure mathematics. And while we can define the Galilean transformation purely mathematically (in fact, I already did), the only reason it's interesting is because of its implications for physics. Physicists like to loot mathematics for useful concepts and methods, and this is no exception.

In some branches of physics, including relativity, the basic building block is an event, which is nothing more that "a thing that happens, sometime and somewhere". Two billiard balls clacking together is an event. Neil Armstrong stepping on the moon is an event. That thing that happens can be nothing -- a particular air molecule sitting in the middle of a room at a given time is also an event, just a rather uninteresting one. A continuous string of events can make up a trajectory. We can think of the path of that billiard ball, or air molecule, as a series of an infinite string of events.

After some consideration, you may see that events are much like points. Events and points both have no "extent" -- they are infinitesimal. Points can be strung together into lines, just like events can be strung into trajectories. So we've decided to model events as points. Points live in manifolds, and we'll call the manifold that contains events spacetime. Coordinates are mapped to a manifold by a coordinate system, which we'll call a reference frame in this context. I'll use these new names from now on; just remember
point = event
line = trajectory
manifold = spacetime
coordinate system = reference frame

One thing we notice in physics is that not all the coordinates are measured in the same way: space is measured with a ruler and time is measured with a clock. While there is no requirement that we do so, we generally choose a coordinate system in which one axis measures time while the other(s) measure space. Since I've decided to restrict us to two-dimensional manifolds, we'll consider only 2D spacetimes with one space and one time dimension. Conventionally, we use x for the space coordinate and t for the time coordinate, so our first reference is the xt system. We'll call our second reference frame the us system -- u is the space coordinate and s the time coordinate.

(Note: the modern convention is to list the time coordinate first; so we'd write (t,x) instead of (x,t). The t axis is drawn vertically, though. For consistency with the previous notation, I will keep the x axis horizontal and write the coordinates at (x,t). Be warned that this is not standard.)

Just as we convert from one coordinate system to another with a coordinate transformation, we convert from one reference frame to anther with a coordinate transformation. The only difference is in interpretation. As a reference frame represents a collection of rulers and clocks, a coordinate transformation tells us how to convert the readings on one set of rulers and clocks to those on another set.

Example: Consider two reference frames that use the same markings on their rulers and same ticks for their clocks. The rulers are offset, though, so that u = 0 corresponds to x = 1.5 (for all time). Moreover, the clocks are not synchronized. s = 0 at t = -0.5 (over all space). Here's a (familiar) picture:
SPtranslation.png
SPtranslation.png (6.65 KiB) Viewed 3984 times

Clearly, (x=2, t=1) = E = (u=0.5, s=1.5). This is an example of a translation coordinate transformation, which we may write as u = x-1.5, s = t+0.5.

Two system in which the axes are parallel, as they are in this case, are called comoving, for reasons that will become apparent shortly.

Exercise 12: Write the coordinate transformation between the xt reference frame, in which distance is measured in meters and time in seconds, and the us frame, which measures distance in kilometers and time in hours. The two systems share an origin and are comoving.
Spoiler:
This is just a scaling of the axes: u = x/1000, s = t/3600.


The Galilean Transformation

The coordinate transformations discussed above keep the axes parallel. The trajectory of an object at constant x would have constant u, and vice versa. Thus, someone at x=0 would see a person at u=0 staying stationary. Therefore, we call these frames comoving. But the example and exercise just covered all types of comoving frames, so let's consider the other case.

First, let's list some observations we make about relatively moving rulers and clocks. These are empirical, so we may well find that they turn out to be wrong after closer examination (he said ominously).
1) All moving clocks stay synchronized with stationary clocks, regardless of their position.
2) Moving rulers have the same lengths as stationary rulers.

Exercise 13: Find the coordinate transformation between the xt and us frames, given that the origins are coincident and the trajectory of u = 0 is given by x = vt. (The same units are used in both frames.)
Spoiler:
Since the t and s clocks remain synchronized, at most we can have a constant offset between them: s = t + b. But since the two frames share the origin, (x=0, t=0) = (u=0, s=0), so b = 0. Similarly, the spatial coordinates must differ only by an offset, although this may depend on time: u = x + a(t). Plugging in u = 0 and x = vt, we get a(t) = -vt. Thus, the coordinate transformation is
u = x - vt
s = t


We call this coordinate transformation the Galilean transformation. (Yes, that's right -- I just derived it in an exercise! I told you that doing exercises is important.) If you've been paying attention, you'll recognize this as a one-axis skew transformation. We can reuse that old image as an example for v = 1/2:
SPskew.png
SPskew.png (7.48 KiB) Viewed 3984 times

Note that, as claimed, u = 0 (the s axis) is given by x = t/2. Note that event E has coordinates (x=2, t=1) and (u=1.5, s=1).

Exercise 14: Verify these coordinates using the Galilean transformation.
Spoiler:
u = x - t/2 = 2 - 1/2 = 1.5; s = t = 1


Well, that's about all I have planned. I hope that everyone who's gone through the examples and exercises has a basic grasp on coordinate systems, coordinate transformations, and the Galilean transformation. I'm happy to answer questions, from Steve or anyone else. But: if you're asking a question about exercise n, I'm going to assume you've done and understand exercises 1 through n-1. If this is not the case, please back up and ask a more basic question.

There's also a lot of other interesting topics we could move on to. I haven't really told you why the Galilean is special in physics. It turns out that Newton's laws are the same in all coordinate frames related by Galilean transformations. Jose and others have mentioned metrics, which are how we measure distances on manifolds. Certain metrics play nicely with certain coordinate transformations -- this leads to the concept of isometries. We can also reexamine those empirical observations about moving clocks and frames. It turns out they're not quite right, in a way that's deeply connected to the isometries of spacetime and the invariance of physical laws.

And a special offer for Steve: After you've gotten comfortable with this derivation of the Galilean transformation, I'd be willing to examine your framework and try to figure out how to translate the Galilean transformation into that language. But only after you've demonstrated some competency with the common language.

And if you're wondering what happened to x': It is more common to use x' and t' instead of u and s for the coordinates of the second reference frame. But this is purely a notational issue.
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Re: Galilean:x' with respect to S'?

Postby steve waterman » Wed Jul 17, 2013 12:53 pm UTC

I am accused on notation ambiguity.
Which is precisely where I find your derivation suffers.

So, let us start at the top, and see if we can agree upon anything at all.

Do we agree
1 that the Galilean given is two Cartesian coincident coordinate systems S(x,y,z) and S'(x'y'z') where
x = x'
y = y'
z = z'
t = t'

2 that at t = 0, S(x,y,z) = S'(x'y'z') ?

3 that at t > 0, S(x,y,z) S'(x'y'z') ?

added - ooops, I am allowed to use system notation, as the Galilean given already employs, so I fixed the above.
Last edited by steve waterman on Wed Jul 17, 2013 1:27 pm UTC, edited 1 time in total.
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Re: Galilean:x' with respect to S'?

Postby JudeMorrigan » Wed Jul 17, 2013 1:27 pm UTC

You know, I don't believe you did the exercises in Schrollini's posts at all, steve. To be blunt, I'm skeptical that you did more than skim the posts at all. That's a pity.

He *did* start at the top. Which is frankly something that *you* are refusing to do. If you there are parts of his derivation where his notation or work appears ambiguous or unclear, I heartily recommend you describing those parts directly. Because going down the road you're desperately trying to drag us down is only going to return us to the Carosel of the Damned. And even more than SexyTalon's mod voice (although that was certainly a factor), my desire to avoid that is why I've been trying to avoid doing more in this thread than encourage you to do your homework.

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Re: Galilean:x' with respect to S'?

Postby ucim » Wed Jul 17, 2013 2:26 pm UTC

steve waterman wrote:Do we agree
1 that the Galilean given is two Cartesian coincident coordinate systems S(x,y,z) and S'(x'y'z') where...

What does it mean for a coordinate system to be "coincident"?

Think about the question, and keep in mind Schrollini's first lesson; that is, separating the idea of a manifold from the idea of a coordinate system, and remembering that points live in a manifold, not in a coordinate system, and also keeping in mind your first Eureka moment.

Draw from the following: Points live in a manifold. A manifold has no structure (like a pile of books). A coordinate system can be imposed on a manifold; the combination will now have some structure. A different coordinate system can be imposed on that same manifold; the combination will be a different combination and have a different structure. Those two structures could be related in some way.

Try your best to answer the question within this framework, without any reference to geometry.

Our shared understanding of this question is essential.

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Re: Galilean:x' with respect to S'?

Postby steve waterman » Wed Jul 17, 2013 2:38 pm UTC

Given S(x',y',z') = S(x,y,z),

S(abscissa,ordinate,applicate) = S'(abscissa,ordinate,applicate)

abscissa x = abscissa x'
ordinate y = ordinate y'
applicate z = applicate z'
t = t'


let's not get confused by insisting upon terms that seem to be ambiguous.
( to at least one camp of thought; distance, metric, etc. etc. etc. )

Any other terms are superfluous/smoke and mirrors/hand-waving as these are the proper mathematical terms.

Hopefully, someone can parse the logic being represented by them...???
Perhaps even try looking up abscissa,ordinate, and applicate (good luck with that, btw)
in several places on the internet, and then parse that together in your head to get your own your
idea/sense/conceptualization of what the words mathematically means.

I am sorry to infuriate those believing I have skimmed over Schrollin's composite posts. I was asked to wait,
so I did. Now we are gonna use new terms like metrics? and new notation like f and u and stuff about cos etc etc etc.

First we agree upon what IS the given.
Then if we get that far, we agree upon what the given means.

Then we both start with the same given and understanding of what that is, THEN I will walk through my specific objections, if I have any, to his derivation. Meanwhile, I am capable of carrying on two conversations at once.
Schollini derivation with only him and I, and my tiny little set of equations above, with others.

For me, the discussion with others must only use mutually accepted terms, otherwise the discussion ( always ) gets silly. If you cannot think in terms of only the three above terms, please do not post comments, now; what I will consider as off-topic. That is, I wish my portion of these two running conversations, to focus upon these 3 terms and corresponding issues. I am not demanding this, I am asking as a favor to try and confine your logic/wording under the umbrella of this limited set of three math terms, please.
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Re: Galilean:x' with respect to S'?

Postby beojan » Wed Jul 17, 2013 2:39 pm UTC

It appears that it was an illusion that Schrollini was getting through to steve at all, and in fact steve was paying only as much attention to Schrollini as he was to the rest of us. At least, if nothing else, Schrollini has produced here a very rigorous introductory course in coordinate geometry, and it may be worth archiving that in the math or science fora, for more honest people to learn from.

Much of the 'misunderstanding', in my opinion, is because of Steve's refusal to answer the simple question asked in the original thread: what is your native language?
So steve, without reducing the attention you can give to Schrollini's final post, and any honest attempt you may make to review his previous posts and actually understand the standard terminology, can you tell us what education you have in the subjects you are currently posting about (maths and physics in general), both formal (courses taken in university or high school), and informal (books read, videos watched, etc).
In addition, what was the language of instruction at each level (what language was the book written in / what language did the professor / teacher speak in)?

--
Later addition (Steve: notice that I'm only adding to my post because nobody had posted after me. Addition made 2013-07-17 14:44 UTC)
The problem with the abscissa, ordinate, applicate terminology is that it locks you into using just one coordinate system. When doing geometric work ("Where does the point at (0,5) end up when reflected in the x-axis?", or "What does a line from (0,6) to (4,7) look like when rotated through 45° clockwise about the point at (0,4)?"), this terminology may be sufficient, but in physics, coordinates are more than just a way to address points. We are interested in looking at the same 'experiment' from more than one point of view, and coordinate transforms enable us to express this mathematically. When working with more than one coordinate system (as we inherently must, when discussing coordinate transforms), the abscissa, ordinate, applicate terminology becomes cumbersome, or even meaningless (look at the drawing giving an example of rotation in the "Several Examples, Chosen with Malice Aforethought" section by Schrollini. In the u-v system, which is the abscissa, and which is the ordinate?)

Edit at 2013-07-17 15:27 UTC to fix typo.
Last edited by beojan on Wed Jul 17, 2013 3:27 pm UTC, edited 1 time in total.

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Re: Galilean:x' with respect to S'?

Postby steve waterman » Wed Jul 17, 2013 2:54 pm UTC

What does it mean for a coordinate system to be "coincident"?

ucim wrote:Think about the question, and keep in mind Schrollini's first lesson; that is, separating the idea of a manifold from the idea of a coordinate system, and remembering that points live in a manifold, not in a coordinate system,

So, does that make it true? The idea, is that I indeed DISAGREE and I start with a different set of initial premises.

given S(x,y,z) and only S(x,y,z)...
Do any points exist "on top of, but not inherent to" ( according to Scrollini ) manifold M?
If we removed manifold M as a thought experiment, what would remain? Would we know exactly where to place S(2,3,4) wrt S(0,0,0)?
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Re: Galilean:x' with respect to S'?

Postby JudeMorrigan » Wed Jul 17, 2013 2:59 pm UTC

beojan wrote:Much of the 'misunderstanding', in my opinion, is because of Steve's refusal to answer the simple question asked in the original thread: what is your native language?

I'm pretty sure he did give a real answer to it, actually. He said he's from Massachusetts, iirc.

ETA: The relevant post:
viewtopic.php?f=2&t=96231&start=720#p3043064

steve waterman wrote:given S(x,y,z) and only S(x,y,z)...
Do any points exist "on top of, but not inherent to" ( according to Scrollini ) manifold M?
If we removed manifold M as a thought experiment, what would remain? Would we know exactly where to place S(2,3,4) wrt S(0,0,0)?

Your question is nonsense. I don't mean that as an insult. I mean it is literally nonsensical.

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Re: Galilean:x' with respect to S'?

Postby beojan » Wed Jul 17, 2013 3:17 pm UTC

JudeMorrigan wrote:
beojan wrote:Much of the 'misunderstanding', in my opinion, is because of Steve's refusal to answer the simple question asked in the original thread: what is your native language?

I'm pretty sure he did give a real answer to it, actually. He said he's from Massachusetts, iirc.

ETA: The relevant post:
http://forums.xkcd.com/viewtopic.php?f= ... 0#p3043064

Huh. I've never noticed that. I'd still like an answer to the rest anyway.

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Re: Galilean:x' with respect to S'?

Postby SecondTalon » Wed Jul 17, 2013 3:35 pm UTC

steve waterman wrote:let's not get confused by insisting upon terms that seem to be ambiguous.
( to at least one camp of thought; distance, metric, etc. etc. etc. )

See, that's the problem. Even as a non-math person, I know that within mathematics and physics and all that jazz, words that seem ambiguous to a layperson like myself are not ambiguous in mathematics. They mean precise, exact things.
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