This is a new thing for me. Among engineers I'm viewed as too theoretical. I'm the guy trying to apply science where empiricism has reigned. Now I feel as if I need to turn that on it's head. So, I may struggle for a time to find the right phrasing. Be patient.
First, I feel it necessary to explain that scientific ideas are of little use to engineers if they don't have a practical application.
Many (not all) engineers use tools without really thinking about why they're using them. So, when I come along with a new technique, they tend to respond: why should I use that? What I'm doing works just fine. They are focused on the end goal - building a machine to sell. If they are only approximating the physics (estimating, guessing, whatever you want to call it), but the result works, that's just fine. It pays the bills.
That may be a bit hyperbolic, but hopefully you get the point. It sets the context for my idea.
I'm tempted to drag you all the way through the journey that led me to where I am, but it's probably not necessary. So, I'll start with two simple statements.
First, what was my goal?
I had success in developing a numerical technique for identifying nonlinear normal modes in rotating machinery. However, it wasn't necessarily easy to use. It's easier to put the machine in a steady state condition to extract linear modes, and then approximate the range of operation as a series of steady state points.
So, my goal was to create a nonlinear analog to linear techniques, so that it would be easier to use. I use two analogies with engineers to explain what I'm trying to do.
First, logarithms. Logarithms transform exponential problems to arithmetic problems:
y = b*x^m
log(y) = m*log(x) + log(b)
y_L = m*x_L + b_L
Second, LaPlace. LaPlace transforms differential problems to algebraic problems.
So, can a nonlinear equation be transformed to allow the use of linear techniques? (I realize that's a vague statement. Several existing techniques could claim they answer that question, but I have to start somewhere, so bear with me)
Second, paradigm changes
I'll state again that the paradigm change I mentioned was not with respect to the essence of the physical world. Rather, it regarded the models used to represent the world. The underlying physics remains basically the same.
There are probably three assumptions engineers make that need to be changed. I'll just state the first one regarding time, since that's what started this all. Then, if someone is interested, we can continue on to discussing the whole thing.
Should I go all the way back to Zeno? Hmm. No, I'll fast forward a bit, though I do think that is where some of our assumptions of time are rooted.
As engineers implement time (though they don't really think about it), it is a comparison to a reference motion. So, they're comparing to cesium cycles even if they don't realize it. And, that comparison is made as a difference. That is the key point. Differences are fundamental to the construction of calculus.
But, as G&K point out, differences are not the only way to make comparisons. For example, comparisons can also be made as ratios. Ratios become fundamental to developing what they call the geometric calculus. (That's not the one I used, but it's good for demonstrating what is changing).
The geometric calculus requires developing a geometric arithmetic. And, consequently, it requires developing a geometric velocity, geometric acceleration, etc.
So, in short, the "differential" velocity is: v_d = (x_2 - x_1) / (t_2 - t_1), where v = velocity, x = displacement, t = time, and (1,2) are different points in time such that x = f(t).
The "geometric" velocity is: v_g = (x_2/x_1) ^ [1/(t_2 - t_1)], which has a more obvious comparison to differential velocity if written as log(v_g) = [log(x_2) - log(x_1)] / (t_2 - t_1).
Edit: I was experimenting with [math] notation. Apparently that doesn't work here.
Hello Resha Caner
Check out this pinned topic on using equations on Physorg forum.
Would like to see where this line of reasoning is headed.
Check out this pinned topic on using equations on Physorg forum.
Would like to see where this line of reasoning is headed.
Firstly, this should probably be called something other than geometric calculus. The natural name given to the calculus of geometric algebra is geometric calculus. This is essentially the extension of classical complex analysis to deal with calculus over an arbitrary Clifford algebra. This is only a small point, mind.
Back to your post: what is your result? At the mo, all I can see is you introducing the operator defined by T(f) = exp[f'(x)/f(x)]. You can't equate this to some physical quantity unless f'/f is dimensionless - so at the mo "geometric velocity" is a bit of a misnomer.
Here are my thoughts on the subject at large: I imagine it's fairly straight forward to produce results analogous to those found in the usual calculus: chain rule, product rule, *anti-derivatives etc - this might be interesting, but I imagine if there is a book on the subject (a quick google of Grossman & Katz suggests there is) then this avenue has been done to death. The main problem I see is that this operation (T) is not a derivation, in the context of differential algebras - so loses the nice properties we associate with derivatives. From an analytical POV, this operator doesn't offer an easy option of attack: our classical function spaces are based on the concept of vector spaces, in which there is no definition of "division". I'm not saying it couldn't be done, but some things would be very hard work.
Perhaps you could elaborate on what your idea actually is?
*Whilst I imagine this doable, I don't think it will be pretty. Obviously integrals of sums = sums of integrals will transform into integral of products into products of integrals etc, but there's there's going to be lots of exponentials flying around that can make things tricky. FTC etc should follow immediately - but this is clear from the definition of T.
Back to your post: what is your result? At the mo, all I can see is you introducing the operator defined by T(f) = exp[f'(x)/f(x)]. You can't equate this to some physical quantity unless f'/f is dimensionless - so at the mo "geometric velocity" is a bit of a misnomer.
Here are my thoughts on the subject at large: I imagine it's fairly straight forward to produce results analogous to those found in the usual calculus: chain rule, product rule, *anti-derivatives etc - this might be interesting, but I imagine if there is a book on the subject (a quick google of Grossman & Katz suggests there is) then this avenue has been done to death. The main problem I see is that this operation (T) is not a derivation, in the context of differential algebras - so loses the nice properties we associate with derivatives. From an analytical POV, this operator doesn't offer an easy option of attack: our classical function spaces are based on the concept of vector spaces, in which there is no definition of "division". I'm not saying it couldn't be done, but some things would be very hard work.
Perhaps you could elaborate on what your idea actually is?
*Whilst I imagine this doable, I don't think it will be pretty. Obviously integrals of sums = sums of integrals will transform into integral of products into products of integrals etc, but there's there's going to be lots of exponentials flying around that can make things tricky. FTC etc should follow immediately - but this is clear from the definition of T.
I didn't show a derivation of Non-Newtonian calculus because G&K have already done so, and do it much more thoroughly than I could. I would just be copying them. Yes, there is a book. I got a copy from a local university to read.
And, yes, they have already handled issues of differentiation, integration, chain rule, etc. I'll say once again that I was applying what they have done, not making further derivations to extend their calculus (see my postscript below).
They eventually develop what they call the star (*) calculus. It comes down to a cookbook for producing a calculus based on any gradient operator one could conceive of.
In my work on machine modes, I would often plot the phase space for the modes (velocity vs. displacement). In linear theory this makes a circle (given that velocity is scaled by the eigenvalue). If the mode is nonlinear, however, the plot is no longer a circle.
Yet, engineers continue to use Fourier transforms on their data, representing the content as a Fourier Series (or, actually, using an FFT tool). The literature is packed with methods for solving nonlinear systems, but the only one I've really ever seen used is to lay out state matrices for the system and integrate them numerically. Often even this is hidden beneath a computer program with a nice interface.
Anyway, I began to wonder if the circle (and the associated functions: sine, cosine, etc.) is really the best basis for nonlinear problems. I found a few papers where people have used non-circular periodic functions, but they are typically numerical rather than symbolic. If that's the case, I might as well use the numerical integration scheme I've already got handy. I wondered if there were a symbolic basis for different periodic functions.
From here the discussion could go several different directions.
In one case, by using a Non-Newtonian calculus, I think I can produce a symbolic solution for certain cases of the Duffing equation. As I've said, it may be that my math is flawed. But if I did it correctly, I think the application provides a powerful tool for solving nonlinear differential equations.
- - -
P.S. It seems you wanted to know the end game, so I tried to say a little more on that. But I also think I should address your comments on developing the calculus further amd making T nondimensional.
My "geometric velocity" (v_g) is not a misnomer as I interpret it. Standard velocity is not dimensionless, but has units of m/s. I think the units of log(v_g) would be log(m)/s. But that's a good question, because it leads to one of the difficulties I have encountered.
In a standard equation for mechanical vibration (mx" + cx' + kx = f), the units on the coefficients are chosen such that their product with the independent variable (x) and its derivatives produces units of force. These coefficients have been given names: mass, damping, stiffness. When you start to dig, you find, for example, that damping is a mathematical convenience that encompasses a wide array of physical phenomena. Regardless, people are attached to the concept of "damping".
I have not worked out all the details on how to handle the coefficients that would multiply v_g. I would rather know if my math is good before I do something like that.
So, let me point out that the "geometric derivative" (if you don't like that name, I'll have to think of another - I just used the same terms as G&K) can be given in terms of the standard derivative through the following algorithm:
DG(f) = exp[ d/dt{ log(f(t)) } ]
where DG is the geometric derivative, d/dt is the standard derivative, and exp is the exponential function.
So, if we choose a function such as f(t) = exp[ sin(wt) ], then
DG1(f) = exp[ w cos(wt) ]
DG2(f) = exp[ -w^2 sin(wt) ]
Which has some interesting similarity to the classic vibration problem.
And, yes, they have already handled issues of differentiation, integration, chain rule, etc. I'll say once again that I was applying what they have done, not making further derivations to extend their calculus (see my postscript below).
They eventually develop what they call the star (*) calculus. It comes down to a cookbook for producing a calculus based on any gradient operator one could conceive of.
In my work on machine modes, I would often plot the phase space for the modes (velocity vs. displacement). In linear theory this makes a circle (given that velocity is scaled by the eigenvalue). If the mode is nonlinear, however, the plot is no longer a circle.
Yet, engineers continue to use Fourier transforms on their data, representing the content as a Fourier Series (or, actually, using an FFT tool). The literature is packed with methods for solving nonlinear systems, but the only one I've really ever seen used is to lay out state matrices for the system and integrate them numerically. Often even this is hidden beneath a computer program with a nice interface.
Anyway, I began to wonder if the circle (and the associated functions: sine, cosine, etc.) is really the best basis for nonlinear problems. I found a few papers where people have used non-circular periodic functions, but they are typically numerical rather than symbolic. If that's the case, I might as well use the numerical integration scheme I've already got handy. I wondered if there were a symbolic basis for different periodic functions.
From here the discussion could go several different directions.
In one case, by using a Non-Newtonian calculus, I think I can produce a symbolic solution for certain cases of the Duffing equation. As I've said, it may be that my math is flawed. But if I did it correctly, I think the application provides a powerful tool for solving nonlinear differential equations.
- - -
P.S. It seems you wanted to know the end game, so I tried to say a little more on that. But I also think I should address your comments on developing the calculus further amd making T nondimensional.
My "geometric velocity" (v_g) is not a misnomer as I interpret it. Standard velocity is not dimensionless, but has units of m/s. I think the units of log(v_g) would be log(m)/s. But that's a good question, because it leads to one of the difficulties I have encountered.
In a standard equation for mechanical vibration (mx" + cx' + kx = f), the units on the coefficients are chosen such that their product with the independent variable (x) and its derivatives produces units of force. These coefficients have been given names: mass, damping, stiffness. When you start to dig, you find, for example, that damping is a mathematical convenience that encompasses a wide array of physical phenomena. Regardless, people are attached to the concept of "damping".
I have not worked out all the details on how to handle the coefficients that would multiply v_g. I would rather know if my math is good before I do something like that.
So, let me point out that the "geometric derivative" (if you don't like that name, I'll have to think of another - I just used the same terms as G&K) can be given in terms of the standard derivative through the following algorithm:
DG(f) = exp[ d/dt{ log(f(t)) } ]
where DG is the geometric derivative, d/dt is the standard derivative, and exp is the exponential function.
So, if we choose a function such as f(t) = exp[ sin(wt) ], then
DG1(f) = exp[ w cos(wt) ]
DG2(f) = exp[ -w^2 sin(wt) ]
Which has some interesting similarity to the classic vibration problem.
QUOTE (Resha Caner+Mar 18 2008, 08:54 PM)
In my work on machine modes, I would often plot the phase space for the modes (velocity vs. displacement). In linear theory this makes a circle (given that velocity is scaled by the eigenvalue). If the mode is nonlinear, however, the plot is no longer a circle.
You might want to look into Liouville integrabiliity and the Arnold-Liouville theorem - this states that if the flow of Liouville integrable system is confined to some compact subset of phase space, then this space is diffeomorphic to the an n-torus. Just nonlinearising doesn't imply you'll move off the n-torus: see KAM theory for lots more detail.
You might want to look into Liouville integrabiliity and the Arnold-Liouville theorem - this states that if the flow of Liouville integrable system is confined to some compact subset of phase space, then this space is diffeomorphic to the an n-torus. Just nonlinearising doesn't imply you'll move off the n-torus: see KAM theory for lots more detail.
QUOTE (Resha Caner+Mar 18 2008, 08:54 PM)
Yet, engineers continue to use Fourier transforms on their data, representing the content as a Fourier Series (or, actually, using an FFT tool).
Sorry - engineers use the Fourier transform to solve nonlinear problems? The only notion of a nonlinear analogue to the Fourier transform I know of is deeply connected with the notion of a Riemann Hilbert problem, and to my knowledge this isn't something widely used by engineers. Perhaps you mean something else?
Sorry - engineers use the Fourier transform to solve nonlinear problems? The only notion of a nonlinear analogue to the Fourier transform I know of is deeply connected with the notion of a Riemann Hilbert problem, and to my knowledge this isn't something widely used by engineers. Perhaps you mean something else?
QUOTE (Resha Caner+Mar 18 2008, 08:54 PM)
Anyway, I began to wonder if the circle (and the associated functions: sine, cosine, etc.) is really the best basis for nonlinear problems. I found a few papers where people have used non-circular periodic functions, but they are typically numerical rather than symbolic. If that's the case, I might as well use the numerical integration scheme I've already got handy. I wondered if there were a symbolic basis for different periodic functions.
I can tell you're an engineer! Things like Fourier series work on a tiny number of problems, and in general there isn't some nice basis of functions you can use to solve a given nonlinear problem. This boils down some results in functional analysis, and the spectra of certain operators. Rudin gives a good introduction if you're interested.
I can tell you're an engineer! Things like Fourier series work on a tiny number of problems, and in general there isn't some nice basis of functions you can use to solve a given nonlinear problem. This boils down some results in functional analysis, and the spectra of certain operators. Rudin gives a good introduction if you're interested.
QUOTE (Resha Caner+Mar 18 2008, 08:54 PM)
In one case, by using a Non-Newtonian calculus, I think I can produce a symbolic solution for certain cases of the Duffing equation. As I've said, it may be that my math is flawed. But if I did it correctly, I think the application provides a powerful tool for solving nonlinear differential equations.
But Duffing's equation involves the standard "Newtonian" derivative - not the T defined above. You'll need to elaborate if I'm to understand what you mean!
But Duffing's equation involves the standard "Newtonian" derivative - not the T defined above. You'll need to elaborate if I'm to understand what you mean!
QUOTE (Resha Caner+Mar 18 2008, 08:54 PM)
My "geometric velocity" (v_g) is not a misnomer as I interpret it. Standard velocity is not dimensionless, but has units of m/s. I think the units of log(v_g) would be log(m)/s.
This makes no sense. I assume you're familiar with dimensional analysis: so you'll realise that you can't add "apples and elephants". Taylor's theorem immediately tells you that you can't make dimensional sense of terms like log(m).
This makes no sense. I assume you're familiar with dimensional analysis: so you'll realise that you can't add "apples and elephants". Taylor's theorem immediately tells you that you can't make dimensional sense of terms like log(m).
QUOTE (Resha Caner+Mar 18 2008, 08:54 PM)
In a standard equation for mechanical vibration (mx" + cx' + kx = f), the units on the coefficients are chosen such that their product with the independent variable (x) and its derivatives produces units of force. These coefficients have been given names: mass, damping, stiffness. When you start to dig, you find, for example, that damping is a mathematical convenience that encompasses a wide array of physical phenomena. Regardless, people are attached to the concept of "damping".
This is not the case - you need to go back over dimensional analysis I think.
This is not the case - you need to go back over dimensional analysis I think.
QUOTE (Resha Caner+Mar 18 2008, 08:54 PM)
DG(f) = exp[ d/dt{ log(f(t)) }]
I think I said as much in my previous post.
I think I need to ask again though: what is your result? It's not currently clear to me!
I think I said as much in my previous post.
I think I need to ask again though: what is your result? It's not currently clear to me!
QUOTE (Euler+Mar 18 2008, 09:24 PM)
Sorry - engineers use the Fourier transform to solve nonlinear problems?
I can tell you're an engineer! Things like Fourier series work on a tiny number of problems, and in general there isn't some nice basis of functions you can use to solve a given nonlinear problem.
I'll get to the serious reply, but first ...
I can tell you're a physicist and not an engineer. It's a "don't judge until you've walked in my shoes" type thing. From the engineer's point of view, physicists are arm chair quarterbacks. Sure, if you've got a view of the whole field and all the time in the world to rewind the tape and play it again, you might think you see a better option. When you're on the field, and in the heat of the moment, it's a different matter.
Didn't I mention that I'm aware the literature is packed with ways to solve nonlinear problems? But no one uses them. Why? Because they typically apply to some obscure, unimportant phenomena. The only reason many were published was to gain someone a PhD.
The cost of searching for the "perfect" method every time a new problem arises is prohibitive. Time after time after time I have seen sophisticated techniques fail and engineering approximations win. Now, if we could apply NASA's budget to every problem, maybe ... or maybe in the future someone will find a better way. But it isn't there yet.
So, engineers aren't claiming to "solve" nonlinear problems in the sense you're thinking. They are using linear techniques to approximate.
In that sense, the FFT dominates structures and vibrations work in mechanical engineering. All other techniques are a trivial pimple on that beastly world. Good Lord, man, you should at least leave the physics building once in awhile and take a walk around the campus ... talk to some engineering profs. Maybe they speak your language better than I do, and can explain it to you.
I can tell you're an engineer! Things like Fourier series work on a tiny number of problems, and in general there isn't some nice basis of functions you can use to solve a given nonlinear problem.
I'll get to the serious reply, but first ...
I can tell you're a physicist and not an engineer. It's a "don't judge until you've walked in my shoes" type thing. From the engineer's point of view, physicists are arm chair quarterbacks. Sure, if you've got a view of the whole field and all the time in the world to rewind the tape and play it again, you might think you see a better option. When you're on the field, and in the heat of the moment, it's a different matter.
Didn't I mention that I'm aware the literature is packed with ways to solve nonlinear problems? But no one uses them. Why? Because they typically apply to some obscure, unimportant phenomena. The only reason many were published was to gain someone a PhD.
The cost of searching for the "perfect" method every time a new problem arises is prohibitive. Time after time after time I have seen sophisticated techniques fail and engineering approximations win. Now, if we could apply NASA's budget to every problem, maybe ... or maybe in the future someone will find a better way. But it isn't there yet.
So, engineers aren't claiming to "solve" nonlinear problems in the sense you're thinking. They are using linear techniques to approximate.
In that sense, the FFT dominates structures and vibrations work in mechanical engineering. All other techniques are a trivial pimple on that beastly world. Good Lord, man, you should at least leave the physics building once in awhile and take a walk around the campus ... talk to some engineering profs. Maybe they speak your language better than I do, and can explain it to you.
QUOTE (Euler+Mar 18 2008, 09:24 PM)
But Duffing's equation involves the standard "Newtonian" derivative - not the T defined above. You'll need to elaborate if I'm to understand what you mean!
This makes no sense. I assume you're familiar with dimensional analysis: so you'll realise that you can't add "apples and elephants". Taylor's theorem immediately tells you that you can't make dimensional sense of terms like log(m).
I'm not sure what I'm trying to do will be clear to you until you have that talk with an engineering prof.
But, I'll say it again. I have two goals.
One: I'm trying to determine if Non-Newtonian calculus could solve specific problems - not all nonlinear problems.
I did ask around at one point if it could be proven that no symbolic solution exists. I didn't think it could be, but I didn't want to overlook the question. So far, I haven't found any such proof, so I see no reason to stop looking (foolish as that may make me appear).
Two: I'm trying to create a nonlinear analogue (specifically for cubic nonlinearities as they have a wide application for me) to the standard linear system.
With respect to the first goal, you may have me with your comment on Duffing's equation. I'll need to think about it some more. I can tell I haven't conveyed my point yet, because I'm not making a simple substitution of derivatives. That's probably because I've been sharing this in pieces. I thought that might help let this settle in.
For example, you seem to be making the hasty judgement about G&K that you chastized me for earlier with regard to the paper I was reading. Were we to be consistent with what you have said, you would need to actually read the book before we continue. If you don't want to do that, maybe we should stop here.
But, I'll give you one more step.
For the linear equation (in a simple form): x" + x = 0
This can also be written as: x' = sqrt(1-x^2)
This can be integrated and solved for x(t) = sin(t)
For a cubic nonlinearity (again, in a simple form): x" + x^3 = 0
This can also be written as: x' = sqrt(1-0.5*x^4)
As far as I know, this can only be handled with an elliptical integral, which does not allow for a solution of x(t).
The next question, then, is if the equation can be transformed into a Non-Newtonian domain and then integrated. Hence, this the step where you lose the association with standard time. The independent variable is no longer time. It's some transformation of time.
You now have me wondering if my transformation is valid.
- - -
As to your second comment, I still think you're not understanding me. Yes, I know what dimensional analysis is, but let me try again.
I put a toothed wheel on a rotating machine, and place two magnetic pickups separated by a known angle to "read" the wheel (I get an electrical pip every time a tooth passes the pickup). This is a common technique on rotating machinery.
So, a tooth passes mag1 and I get pip1. I'll call that displacement = 0. When that same tooth passes mag2, I get pip2 and I know the rotational displacement of the wheel at that point in time (because I'm also running a clock).
So, now I have x_1, x_2, t_1, and t_2. What I do with that data from this point is up to me. I can color it blue if I want as long as that is meaningful to the problem I'm working on.
The standard approach would be to say: velocity = (x_2 - x_1) / (t_2 - t_1).
An equally valid math function is: (x_2 / x_1) ^ (1 / (t_2 - t_1)).
But is that meaningful to the problem? Maybe. I think you're trying to state a definitive "no", while I'm still at maybe.
For example, what if my system is described by y = x^2 (some hydraulic systems are). It's sure a heck of lot easier to draw a straight line on log-log paper than to plot out the points of a parabola. I know this is a lost art given computers, and no one uses slide rules any more either. But solving a problem on a slide rule or by using log-log paper didn't invalidate it because of an issue with dimensions.
So, I have a list of discrete values, x. And I have a list of discrete values, t. If I can produce an equation that consistently defines the relationship between these two lists of values, and it can be symbolically solved, I think that is very powerful.
I think I am close to doing that for the cubic nonlinearity. But I'm not done, and so far no one but poor ol' me seems to think it's worth trying. So, if you can show me a definite roadblock, I'll enrole myself in a recovery program and try to get over it.
This makes no sense. I assume you're familiar with dimensional analysis: so you'll realise that you can't add "apples and elephants". Taylor's theorem immediately tells you that you can't make dimensional sense of terms like log(m).
I'm not sure what I'm trying to do will be clear to you until you have that talk with an engineering prof.
But, I'll say it again. I have two goals.
One: I'm trying to determine if Non-Newtonian calculus could solve specific problems - not all nonlinear problems.
I did ask around at one point if it could be proven that no symbolic solution exists. I didn't think it could be, but I didn't want to overlook the question. So far, I haven't found any such proof, so I see no reason to stop looking (foolish as that may make me appear).
Two: I'm trying to create a nonlinear analogue (specifically for cubic nonlinearities as they have a wide application for me) to the standard linear system.
With respect to the first goal, you may have me with your comment on Duffing's equation. I'll need to think about it some more. I can tell I haven't conveyed my point yet, because I'm not making a simple substitution of derivatives. That's probably because I've been sharing this in pieces. I thought that might help let this settle in.
For example, you seem to be making the hasty judgement about G&K that you chastized me for earlier with regard to the paper I was reading. Were we to be consistent with what you have said, you would need to actually read the book before we continue. If you don't want to do that, maybe we should stop here.
But, I'll give you one more step.
For the linear equation (in a simple form): x" + x = 0
This can also be written as: x' = sqrt(1-x^2)
This can be integrated and solved for x(t) = sin(t)
For a cubic nonlinearity (again, in a simple form): x" + x^3 = 0
This can also be written as: x' = sqrt(1-0.5*x^4)
As far as I know, this can only be handled with an elliptical integral, which does not allow for a solution of x(t).
The next question, then, is if the equation can be transformed into a Non-Newtonian domain and then integrated. Hence, this the step where you lose the association with standard time. The independent variable is no longer time. It's some transformation of time.
You now have me wondering if my transformation is valid.
- - -
As to your second comment, I still think you're not understanding me. Yes, I know what dimensional analysis is, but let me try again.
I put a toothed wheel on a rotating machine, and place two magnetic pickups separated by a known angle to "read" the wheel (I get an electrical pip every time a tooth passes the pickup). This is a common technique on rotating machinery.
So, a tooth passes mag1 and I get pip1. I'll call that displacement = 0. When that same tooth passes mag2, I get pip2 and I know the rotational displacement of the wheel at that point in time (because I'm also running a clock).
So, now I have x_1, x_2, t_1, and t_2. What I do with that data from this point is up to me. I can color it blue if I want as long as that is meaningful to the problem I'm working on.
The standard approach would be to say: velocity = (x_2 - x_1) / (t_2 - t_1).
An equally valid math function is: (x_2 / x_1) ^ (1 / (t_2 - t_1)).
But is that meaningful to the problem? Maybe. I think you're trying to state a definitive "no", while I'm still at maybe.
For example, what if my system is described by y = x^2 (some hydraulic systems are). It's sure a heck of lot easier to draw a straight line on log-log paper than to plot out the points of a parabola. I know this is a lost art given computers, and no one uses slide rules any more either. But solving a problem on a slide rule or by using log-log paper didn't invalidate it because of an issue with dimensions.
So, I have a list of discrete values, x. And I have a list of discrete values, t. If I can produce an equation that consistently defines the relationship between these two lists of values, and it can be symbolically solved, I think that is very powerful.
I think I am close to doing that for the cubic nonlinearity. But I'm not done, and so far no one but poor ol' me seems to think it's worth trying. So, if you can show me a definite roadblock, I'll enrole myself in a recovery program and try to get over it.
QUOTE (Resha Caner+Mar 19 2008, 01:23 AM)
I can tell you're a physicist and not an engineer.
I'm not a physicist.
I'm not a physicist.
QUOTE (Resha Caner+Mar 19 2008, 01:23 AM)
Didn't I mention that I'm aware the literature is packed with ways to solve nonlinear problems? But no one uses them. Why? Because they typically apply to some obscure, unimportant phenomena. The only reason many were published was to gain someone a PhD.
I tend not to judge the merit of a mathematical result based on whether or not engineers choose to use them.
I tend not to judge the merit of a mathematical result based on whether or not engineers choose to use them.
QUOTE (Resha Caner+Mar 19 2008, 01:23 AM)
So, engineers aren't claiming to "solve" nonlinear problems in the sense you're thinking. They are using linear techniques to approximate.
So engineers aren't using the Fourier transform to solve nonlinear problems.
So engineers aren't using the Fourier transform to solve nonlinear problems.
QUOTE (Resha Caner+Mar 19 2008, 02:23 AM)
I can tell you're a physicist and not an engineer. It's a "don't judge until you've walked in my shoes" type thing. From the engineer's point of view, physicists are arm chair quarterbacks.
Aren't you the guy who reads string theory papers but doesn't even know what 'the bulk' is? Talk about hypocrisy!
And if you couldn't tell, Euler's knowledge of mathematics is many orders of magnitude better than yours. He's moped the floor with you in this thread.
And those methods most certainly are used. Do you know who employs my father for the most external contract work? Lockheed, BAE and Airbus. Three companies with staggering amounts of resources and who make use of non-linear systems all the time. You do realise that the Navier-Stokes equations, which describe fluid flow, are non-linear right?Right?
Last week I finished reviewing, for a journal, a paper on non-linear quantum systems (ie the equation which describes super conductivity). Another area with enormous applications to things like quantum computing and condensed matter physics.
So do us a favour and stop trying to pretend you've got the answers to all this. When it comes to mathematics, Euler will mop the floor with you. If it's physics, a few of us will pitch in. And by the looks of it you think you're an engineer and trying to go down the "You think about physics, we do physics!" route but you managed to fall flat on your face with that one too.
Aren't you the guy who reads string theory papers but doesn't even know what 'the bulk' is? Talk about hypocrisy!
And if you couldn't tell, Euler's knowledge of mathematics is many orders of magnitude better than yours. He's moped the floor with you in this thread.
QUOTE
Didn't I mention that I'm aware the literature is packed with ways to solve nonlinear problems? But no one uses them. Why? Because they typically apply to some obscure, unimportant phenomena. The only reason many were published was to gain someone a PhD.
Firstly, Euler is a PhD in non-linear systems. Secondly, my father is an extremely widely published professor of non-linear systems and has been head of various engineering departments and groups in universities. And those methods most certainly are used. Do you know who employs my father for the most external contract work? Lockheed, BAE and Airbus. Three companies with staggering amounts of resources and who make use of non-linear systems all the time. You do realise that the Navier-Stokes equations, which describe fluid flow, are non-linear right?Right?
Last week I finished reviewing, for a journal, a paper on non-linear quantum systems (ie the equation which describes super conductivity). Another area with enormous applications to things like quantum computing and condensed matter physics.
So do us a favour and stop trying to pretend you've got the answers to all this. When it comes to mathematics, Euler will mop the floor with you. If it's physics, a few of us will pitch in. And by the looks of it you think you're an engineer and trying to go down the "You think about physics, we do physics!" route but you managed to fall flat on your face with that one too.
QUOTE (Resha Caner+Mar 19 2008, 02:11 AM)
I'm trying to determine if Non-Newtonian calculus could solve specific problems - not all nonlinear problems.
So you don't have a result - rather a question?
So you don't have a result - rather a question?
QUOTE (Resha Caner+Mar 19 2008, 02:11 AM)
I did ask around at one point if it could be proven that no symbolic solution exists. I didn't think it could be, but I didn't want to overlook the question. So far, I haven't found any such proof, so I see no reason to stop looking (foolish as that may make me appear).
See differential Galois theory, and more concretely in the case of second order problems, the Painlevé transcendents.
See differential Galois theory, and more concretely in the case of second order problems, the Painlevé transcendents.
QUOTE (Resha Caner+Mar 19 2008, 02:11 AM)
I'm trying to create a nonlinear analogue (specifically for cubic nonlinearities as they have a wide application for me) to the standard linear system.
You're trying to create a nonlinear analogue of a linear system? This isn't clear.
You're trying to create a nonlinear analogue of a linear system? This isn't clear.
QUOTE (Resha Caner+Mar 19 2008, 02:11 AM)
With respect to the first goal, you may have me with your comment on Duffing's equation. I'll need to think about it some more. I can tell I haven't conveyed my point yet, because I'm not making a simple substitution of derivatives. That's probably because I've been sharing this in pieces. I thought that might help let this settle in.
At the moment, I'm not sure if you have a point. From what I understand, you've essentially seen this operator T in a book, and wonder if it could be used to solve things like Duffing's equation.
At the moment, I'm not sure if you have a point. From what I understand, you've essentially seen this operator T in a book, and wonder if it could be used to solve things like Duffing's equation.
QUOTE (Resha Caner+Mar 19 2008, 02:11 AM)
For example, you seem to be making the hasty judgement about G&K that you chastized me for earlier with regard to the paper I was reading. Were we to be consistent with what you have said, you would need to actually read the book before we continue. If you don't want to do that, maybe we should stop here.
This would only hold any water if I didn't understand the mathematics you're referring to. I'll let you know when it gets too complicated for me.
This would only hold any water if I didn't understand the mathematics you're referring to. I'll let you know when it gets too complicated for me.
QUOTE (Resha Caner+Mar 19 2008, 02:11 AM)
For the linear equation (in a simple form): x" + x = 0
This can also be written as: x' = sqrt(1-x^2)
This can be integrated and solved for x(t) = sin(t)
For a cubic nonlinearity (again, in a simple form): x" + x^3 = 0
This can also be written as: x' = sqrt(1-0.5*x^4)
As far as I know, this can only be handled with an elliptical integral, which does not allow for a solution of x(t).
In both of these, your "1" should be replaced with an arbitrary constant, but no matter.
This can also be written as: x' = sqrt(1-x^2)
This can be integrated and solved for x(t) = sin(t)
For a cubic nonlinearity (again, in a simple form): x" + x^3 = 0
This can also be written as: x' = sqrt(1-0.5*x^4)
As far as I know, this can only be handled with an elliptical integral, which does not allow for a solution of x(t).
In both of these, your "1" should be replaced with an arbitrary constant, but no matter.
QUOTE (Resha Caner+Mar 19 2008, 02:11 AM)
As far as I know, this can only be handled with an elliptical integral, which does not allow for a solution of x(t).
Of course it does - the solution just happens to be in the form of an elliptic integral. Just because you don't happen to like them, or find them complicated doesn't make the solution any less valid. If you are hoping for a different solution, then you need to learn some basic analysis - specifically Picard's theorem. If you are hoping to turn an elliptic integral into something you understand more, please refer to differential Galois theory (again).
Of course it does - the solution just happens to be in the form of an elliptic integral. Just because you don't happen to like them, or find them complicated doesn't make the solution any less valid. If you are hoping for a different solution, then you need to learn some basic analysis - specifically Picard's theorem. If you are hoping to turn an elliptic integral into something you understand more, please refer to differential Galois theory (again).
QUOTE (Resha Caner+Mar 19 2008, 02:11 AM)
Yes, I know what dimensional analysis is, but let me try again.
If you know what dimensional analysis is, then you'll understand that you can't make dimensional sense of terms like log(m). Your problem would need to be such that f'/f was dimensionless - as I said earlier.
I would recommend you look at some of the common ways to deal with nonlinear PDEs. As a starter perhaps look into Hamiltonian dynamics and Liouville integrability theorems. After which you could start to look at inverse scattering approaches, the Lax formalism, Lie's method of prolongation etc. Although before this I would recommend you go over some of the results in analysis that I've outlined for you.
If you know what dimensional analysis is, then you'll understand that you can't make dimensional sense of terms like log(m). Your problem would need to be such that f'/f was dimensionless - as I said earlier.
I would recommend you look at some of the common ways to deal with nonlinear PDEs. As a starter perhaps look into Hamiltonian dynamics and Liouville integrability theorems. After which you could start to look at inverse scattering approaches, the Lax formalism, Lie's method of prolongation etc. Although before this I would recommend you go over some of the results in analysis that I've outlined for you.
QUOTE (AlphaNumeric+Mar 19 2008, 08:00 AM)
And if you couldn't tell, Euler's knowledge of mathematics is many orders of magnitude better than yours. He's moped the floor with you in this thread.
Firstly, Euler is a PhD in non-linear systems. Secondly, my father is an extremely widely published professor of non-linear systems and has been head of various engineering departments and groups in universities.
And those methods most certainly are used. Do you know who employs my father for the most external contract work? Lockheed, BAE and Airbus. Three companies with staggering amounts of resources and who make use of non-linear systems all the time. You do realise that the Navier-Stokes equations, which describe fluid flow, are non-linear right?Right?
Have you contributed anything productive yet? Because your dad is an expert that makes you one? Maybe you guys should sign up for the new TV show: "My Dad's Better Than Your Dad".
This isn't a contest to show who's smarter - or at least I didn't intend it to go that direction.
All your comment did was reinforce to me that you don't know what you're talking about. All the references you mentioned have little to nothing to do with what I was talking about.
I've interacted with engineers from the aerospace companies. I've dealt with "consultants". Of course I don't know your dad, so he may be an exception, but my experience and the experience of the aerospace engineers I know has not been good. "Consultants" are typically hired by overzealous managers who are looking for a quick fix. The consultant comes in and does a good job of impressing the manager by talking a lot and theorizing about how the problem might be solved. Then the consultant leaves, the problem still isn't fixed, and when the engineers finally figure it out, the "consultant" gets all the credit.
Again, it's the same arrogance of which I was accused (and for which I apologized), that people think they're smart enough that they can forego the background requirements and solve a problem just because they have a diploma.
Firstly, Euler is a PhD in non-linear systems. Secondly, my father is an extremely widely published professor of non-linear systems and has been head of various engineering departments and groups in universities.
And those methods most certainly are used. Do you know who employs my father for the most external contract work? Lockheed, BAE and Airbus. Three companies with staggering amounts of resources and who make use of non-linear systems all the time. You do realise that the Navier-Stokes equations, which describe fluid flow, are non-linear right?Right?
Have you contributed anything productive yet? Because your dad is an expert that makes you one? Maybe you guys should sign up for the new TV show: "My Dad's Better Than Your Dad".
This isn't a contest to show who's smarter - or at least I didn't intend it to go that direction.
All your comment did was reinforce to me that you don't know what you're talking about. All the references you mentioned have little to nothing to do with what I was talking about.
I've interacted with engineers from the aerospace companies. I've dealt with "consultants". Of course I don't know your dad, so he may be an exception, but my experience and the experience of the aerospace engineers I know has not been good. "Consultants" are typically hired by overzealous managers who are looking for a quick fix. The consultant comes in and does a good job of impressing the manager by talking a lot and theorizing about how the problem might be solved. Then the consultant leaves, the problem still isn't fixed, and when the engineers finally figure it out, the "consultant" gets all the credit.
Again, it's the same arrogance of which I was accused (and for which I apologized), that people think they're smart enough that they can forego the background requirements and solve a problem just because they have a diploma.
QUOTE (Euler+Mar 19 2008, 08:01 AM)
So you don't have a result - rather a question?
I said I had an idea. Everything starts with an idea, but considering your crack about being an "ideas man", I tried to make it clear that I realize it needs to be backed up with details. I knew that long ago. I've been pursuing the details, and never claimed I was ready to publish.
If I was ready to publish, that's what I would do. I wouldn't be talking about it in an Internet forum.
Yes, at one point I thought I had a result, but my acquaintances pointed out some holes. That all happened before I ever got here. So I've been looking to see if I can add some rigor to what I've done. I'm trying to determine if added rigor gets me back to the result I would like to see, or if it proves my approach unviable.
I've only posted as much as I have because some seemed curious. But, if I have to start shouting over the background noise, it's not worth it.
I will investigate your suggestions. But I'm not convinced they were sincere. Why? Because you continue to talk down to me. It feels more as if you're testing me than trying to help me. As an example:
Yes, I passed my college calculus class. I know how to integrate. I could give you initial conditions that would make the constant 1, but I thought I made it clear I was over-simplifying the equations to make a quick point.
Maybe you are frustrated, but stuff like that frustrates me as well.
If I have misjudged you, then I'll apologize. I think you have misjudged me as well. As I said, I'll think some more on your comments and see where they lead me. Given that your opinion seems formed on what I have done, I don't see anything I would say from this point forward would be productive for either of us.
I said I had an idea. Everything starts with an idea, but considering your crack about being an "ideas man", I tried to make it clear that I realize it needs to be backed up with details. I knew that long ago. I've been pursuing the details, and never claimed I was ready to publish.
If I was ready to publish, that's what I would do. I wouldn't be talking about it in an Internet forum.
Yes, at one point I thought I had a result, but my acquaintances pointed out some holes. That all happened before I ever got here. So I've been looking to see if I can add some rigor to what I've done. I'm trying to determine if added rigor gets me back to the result I would like to see, or if it proves my approach unviable.
I've only posted as much as I have because some seemed curious. But, if I have to start shouting over the background noise, it's not worth it.
I will investigate your suggestions. But I'm not convinced they were sincere. Why? Because you continue to talk down to me. It feels more as if you're testing me than trying to help me. As an example:
QUOTE
In both of these, your "1" should be replaced with an arbitrary constant, but no matter.
Yes, I passed my college calculus class. I know how to integrate. I could give you initial conditions that would make the constant 1, but I thought I made it clear I was over-simplifying the equations to make a quick point.
Maybe you are frustrated, but stuff like that frustrates me as well.
If I have misjudged you, then I'll apologize. I think you have misjudged me as well. As I said, I'll think some more on your comments and see where they lead me. Given that your opinion seems formed on what I have done, I don't see anything I would say from this point forward would be productive for either of us.
You said things like:
which suggest you had implemented the idea in some way. Hence my continued requests for elaboration!
QUOTE (Resha Caner+Mar 18 2008, 08:54 PM)
- I'll say once again that I was applying what they have done.
- I think I can produce a symbolic solution for certain cases of the Duffing equation.
which suggest you had implemented the idea in some way. Hence my continued requests for elaboration!
QUOTE (Resha Caner+Mar 19 2008, 01:57 PM)
I will investigate your suggestions.
How kind of you!
How kind of you!
QUOTE (Resha Caner+Mar 19 2008, 01:57 PM)
Given that your opinion seems formed on what I have done, I don't see anything I would say from this point forward would be productive for either of us.
A thank you would help.
A thank you would help.
MR.CANER -the solutions to the equations of the GTR that are nonlinear,gives us two resulted:discrete and continue solutions.then the spacetime is very much more complex that imagine,so as the geometry that study motion,displacement and time.Is this?
caner-your idea to connect geometric and algebrics motions,through the log and exp,appear me very intelligent to explain the motion,in the curvatures,that are deformations of spacetime continuos.see wheeler,the geometry-foam.
the universe might be seen as multiples mirrors as symmetries fundamental to the
groupf of lie,through the painleve differential groups,that implies the transcendence
as rational functions obtained by the algebric equations of degrees upper 5.
groupf of lie,through the painleve differential groups,that implies the transcendence
as rational functions obtained by the algebric equations of degrees upper 5.
I think that the chirality of asymmetric spinor fields define the arrow of time.being so the spacetime continuos are closed curvatures.Being the time splitted by two coordinates,of opposed directions.thence will calcule,through the asymmetry of
left-handed and right-handed spinors,the time dilatation and the space contraction,derived of the origin constant of speed of light,that is generated by the
breakdown of symmetry pt,restaured by nonlinear lorentz transformations
left-handed and right-handed spinors,the time dilatation and the space contraction,derived of the origin constant of speed of light,that is generated by the
breakdown of symmetry pt,restaured by nonlinear lorentz transformations
i awaiting response
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