A reply to “Angular Velocity and Angular Momentum”

In Angular Velocity and Angular Momentum Miles claims the current definitions of angular velocity and angular momentum are wrong.

To start with, look again at the basic equations

p = mv

L = rmv

Where L is the angular momentum. This equation tells us we can multiply a linear momentum by a radius and achieve an angular momentum. Is that sensible? No. It implies a big problem of scaling, for example. If r is greater than 1, the effective angular velocity is greater than the effective linear velocity. If r is less than 1, the effective angular velocity is less than the effective linear velocity. How is that logical?

First of all momentum and angular momentum are vectors and the correct definitions should read \vec{p}=m \vec{v} and \vec{L}=\vec{r} \times \vec{p}. Miles took the absolute value of the angular momentum |\vec{L}|=|\vec{r}| |\vec{p}| \sin(\vec{r},\vec{p}) while tacitly assuming that \vec{r}\bot \vec{p} (as in circular motion). It escapes us how he constructs his “problem of scaling”, but it seems he didn’t notice that \vec{L} and \vec{p} are different quantities. We now go straight to the part of the paper where a serious error occurs.

The correction for all this is fairly simple, although it required me to study the Principia very closely. We need a new equation to go from tangential or linear velocity to ω. Newton does not give us that equation, and no one else has supplied it since then. We can find it by following Newton to his ultimate interval, which is the same as going to the limit. We use the Pythagorean Theorem. As t→0,

ω2 → v2 – Δv2
and, v2 + r2 = (Δv + r)2
So, by substitution, ω2 + Δv2 + r2 = Δv2 + 2Δvr + r2
Δv = √ v2 + r2) – r = ω2/2r

ω = √[2r√v2 + r2) – 2r2]
r = √[ω4/(4v2 – 4ω2)]

This is Miles’ replacement for the current defintion of angular velocity. Apparently he got \omega^2 \to v^2 - \Delta v^2 and v^2 + r^2 = (\Delta v + r)^2 from Newton’s principia. It suffices to say that the units in the last equation do not match and it is therefore wrong.

Update:
Some days ago Miles added a section where he is supposedly clarifying the problems with units. He argues that he simply redefined \omega to be an acceleration. Redefine he did (according to the highlighted text), but \omega is still not an acceleration. It is a quantity which has no physical meaning, because the squares of two different quantities are added.

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11 Responses to A reply to “Angular Velocity and Angular Momentum”

  1. Steven Oostdijk says:

    So, that is your way out of this? A quantity with “no physical meaning”….? 🙂

    I think you should at least start to reread Mathis to find out that v and r have the same units in circular motion.

  2. Cee says:

    Dear Steven,

    Would you care to point out where Mathis writes this explicitly? In his 2010 Addendum he writes explicitly that “v = 1/s^2” (in words: velocity has dimensions one over second squared). He then argues that since time and length are “inverse parameters”, this would be dimensionally the same as length/time (e.g. meters/second), i.e. it really is a velocity.

    For units in the bolded equations to work out r, v and omega all have to have to same dimensions (this is the crucial argument, please take some time to think this through). If we take the quote “v = 1/s^2” seriously, r and omega must thus have dimension one over time squared. But r is a length which, even following Mathis’ arguments, has dimensions one over time.

    Furthermore, he derives the equation “v = omega/r” as a limiting case. If omega and r both had the same units, this equations implies that v has none.

    Judging from other comments you have posted on several other sites it seems that you’re quite an active supporter of Mathis’ work. Would you be interested in discussing the equations in this paper in detail with us and subjecting them to experiment?

    Sincerely

  3. Steven Oostdijk says:

    Mathis explains it in his paper on “What is Pi?”: http://www.milesmathis.com/pi.html.

    ” It means that the radius is a velocity itself. Go back to the Cartesian graph and you will remember that any straight line in the graph is a constant velocity. You know, the slope, the intercept, all that? Well, our radius is somewhat like that. Instead of writing r, we should write r/t. The radius is r/t. When we start comparing r to the circumference, we have to assume that the circumference is drawn with the same velocity. If we are going to ignore it later, as the geometry does, we have to assume that it is equal. So let’s do that.

    Axiom 1: the velocity of the radius is equal to the velocity of the circumference.”

    He also explains that Newton and others use this same assumption in his paper on his correction to “a=v^2/r”.

    “If v does not change as t changes, then v is a constant. The derivative of a constant is zero. Therefore it makes no sense to differentiate a constant velocity, even if it happens to be labeled Δv.
    You may say, “OK, but is all that legal? Can you combine different vectors in a vector addition? Isn’t there some rule about mixing acceleration vectors and velocity vectors?” Yes, there are rules. The length of the vector stands only for its numerical value: that’s why you must keep careful track of angles. But no one has ever had any problem with the way that distance vectors and velocity vectors were combined in this problem, historically. The radius of the circle is obviously not a velocity; it is a distance. But both the textbook and Feynman use the radius and the velocity vectors as values that can be put in the same equation. If you can do that, why not use acceleration vectors as well? The answer is, you can, and Newton, the textbook and Feynman all do that, too. They just don’t call attention to it. They solve this problem without ever defining their variables. ”

    I’m a supporter of Mathis work but that is mostly a hobby, I do not plan to spend more time on it than I do now already. Also I’m trying to work towards undeniable experiments with Mathis at the moment though that still could take some time. I think it will have to come from an update of Signal Processing theory since that can be verified with electronic circuit measurements.

  4. Steven Oostdijk says:

    Apparently one cannot edit post later. Here is link I forgot to the “a=v^2/r” paper:

    http://www.milesmathis.com/avr.html

  5. D says:

    I believe we can all agree that any mathematical system must be self consistent to be of any value. In that spirit I will demonstrate where Mathis contradicts his own ideas. If one reads his paper on the differentiation of the natural logarithm you will find he takes exception to taking limits as a variable goes to zero. His entire argument in that paper is that since the number 1 is the unit lenght, and according to him in the definition of derivative it should have h->1 not h->0. Apparently limit as h->0 is not a valid operation since 1 is the ‘smallest unit on the number line’. Now I in no way concede that point, h->0 IS a perfecly valid operation. The internal contradiction occurs in the above Where he says “we use the Pythagoren Theorem. As t->O…”. It seems MM picks and chooses how limits can work within his framework to suit his needs at any given time. Even If one assumes that all his other work in both papers is correct, which I don’t, then the this fact alone demonstrates how either his work with the logarithm cannot hold or this paper cannot hold (in fact, neither holds).

  6. D says:

    Did I read that correctly, you are going to correct signal processing theories. I’ll tell you what, when even 1 of your corrections to a theory can be used to digitally encode audio signals, improve ANY aspect of the telecommunications system, or build a better radio telescope, then I will be the first person to acknowledge the success. Do you think it is dumb luck that the telecommunications revolution has happened when it appears you seem to think they are working with broken tools?

    Is there even one aspect of physics/math that Mathis cannot improve on? Even Newton and Gauss had limit’s to the breadth of material they worked on. Are you so vain as to think the last 300 years of advances are all wrong and that one man can alone rewrite what took humanity millenia to achieve?

    • Steven Oostdijk says:

      These kind of threads always reveal more about the human psyche than about physical or math theory.

      Nowhere on his site does Mathis claim that current theory is fully wrong. He just argues/shows it is a collection of glorified heuristics and fluffy math with very little consistency in the theoretical basics. He is working on those basics and even then just scratching the surface.

      About signal processing theories, I picked that because it is a field where the theory can be easily verified by (electronic) measurements and it is my own area of expertise. Just study the effectivity of biquad filtering, which are based on simple differences, and compare that to Mathis theories.

      Steven

      • Gus Mueller says:

        ”Steven Oostdijk says:
        October 31, 2010 at 7:09 am
        These kind of threads always reveal more about the human psyche than about physical or math theory.”

        Did you really just say that, Miles? True of your work, as well, I should say. And we’re not even discussing your Humbert Humbert-like fascination with the little blonde girl. If Taschen wouldn’t publish it, it either wasn’t worth publishing or was beyond disturbing.

  7. D says:

    Steven, I must disagree. The sheer volume and breadth of his writings seem to suggest that he finds fault in much of current mathematics. Prime examples are his arguments against the derivative of the natural logarithm and the accepted value of pi. These conepts are so fundamental to such a vast amount of mathematics that it’s safe to say everything would crumble if he were correct.

    I assure you there is no “fluffy math” in real or complex analysis, number theory, or mathematical logic. It is foolish to think all of math can be discussed when your only tools are a small amount of calculus and some basic algebra. There is absolutely not even one theorem accepted as true in any of those fields that is accepted based on heuristics, not one!

    His notions of circumference being a velocity and the time taken to draw it are also rubish. He clearly lacks the ability to abstract away from the act of drawing a circle to the notion of a locus of points satisfying an equation.

    He seems to feel all mathematical concepts must have physical manifestations. What is a physical manifestation of a non commutative algebra, or matrix, or a diffeomorphism, or a function for that matter?

    Long ago mathematics was taken off the questionable foundation of number as a length or something constructible with compas and straight edge, instead it was given a firm basis in axiomatic set theory.

    I ask you, why does a circle with radius 1 need time anywhere in it’s description. Consider the set of all (x , y) satisfying x^2 y^2 = 1, how is a velocity or acceleration or anything related to either needed? The circle is well defined and unambiguous the way I described it, nothing need be added.

    In my opinion for “papers” on mathematics they are very lite on the math. There seems to be hoarded of I’ll defined concepts, lots of handwaving, and even more self praise. If you were to look at a real math journal, say annals of mathematics, you would see well defined terms, Clear and unambiguousl statements, the minimum amount of verbal arguments and more symbolic reasoning, and no self praise.

    If you can show me any fluffy math in say Principles of Mathematical Analysis by Rudin I will admit it. That text is the classic in introductory real analysis and a few generation of mathematicians learned from it. It gives a construction of the reals beginning with only the naturals, covers basic topology, some functional analysis, and cocludes with measure theory and the Lebesgue integral. I challenge you to demonstrate one instance of fluffy math. Since it covets foundations up to some fairly sophisticated ideas, it should be easy if what you say is true.

  8. D says:

    I apologize for typos. My iPod OS has an autocomplete feature for typing and occasionally the word it chooses is not the one I wanted, ie: “…hoarded of ill…”. It’s a hassle to even get ill, it keeps changing it it I’ll.

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