Wednesday, April 16, 2014

Parabolic Range/Height Elliptical Envelope Results of the Day

We're still working on figuring out a way to geometrically, or verbally, explain the fact that an ellipse is formed by by tracing through the apexes of a family of parabolas that describe the trajectories of a projectile launched with the same initial velocity but different launch angles.  Try saying that three times fast, and in the meantime, see picture 1, and refer to the excellent open access article that derives the ellipse mathematically[1].

After I wrote about the elliptical envelope a few days ago +rocktoasted asked if there was a way to describe the construction of the ellipse without using equations or algebra.  Hence, the search for a geometrical explanation was born.  +Bruce Elliott joined in on the work, and now we have a new collaborator, the only already PhD'ed physicist in the house, my wife Elaine.  If you'd like to get in on the collaboration by contributing observations that lead to explaining how to construct the ellipse in [1], please do! Maybe in the end we can right the whole thing up and submit it to one of the +Mathematical Association of America's journals.  It might just be the first ad-hoc internet math collaboration to publish there.  Who knows??? But, I digress :)

We played around with the diagram of how to make a parabola from the intersection of a plane and set of cones, and came up with the following observations.

Elaine intuited that the base circle shown in picture 2 was actually traced out by the line representing the ground for each parabolic trajectory.

Using the equations from the paper [1], we saw two mathy things immediately.  First, if you put 45 degrees, (the angle that gives the maximum range), into the x equation, and 90 degrees, (the angle that gives the maximum height) into the y equation, you'll find that the maximum range is always twice the maximum possible height of the projectile at a given fixed velocity.  Bruce saw this first and mentioned it in respect to the eccentricity of the ellipse in the paper always being equal to one half.

The second thing we found requires a bit of calculus.  The x range is equal to half of the derivative with respect to theta of the apex height.

I'm not entirely sure what the significance of this yet, but it feels kind of nice from a circular perspective.  The x coordinate of a point on a circle is equal to the derivative of the y coordinate of a circle, (picture 4).  Notice that the patch factor for the ellipse is just one over the eccentricity, (the ratio of the maximum y distance vs. the maximum x distance of the ellipse).

OK, so so far, we have a family of parabolas formed by laying a plane parallel and tangent to a cone and then pushing that plane through the cone and assuming that the ground is perpendicular to the line that drops straight down from the apex of the cone.  Here are the geometry-ish statements that go with this system.

1.  When the plane is just tangent to the cone, the parabola that's formed is actually a straight line, (line CK in the diagrams) and represents the trajectory of the projectile if you launched it straight up at a launch angle of 90 degrees.

2.  The height of the line CK is proportional to the kinetic energy of the particle at launch, or the potential energy of the particle at the apex, (they're the same for the 90 degree launch angle represented by line CK).

3.  As the launch angle sweeps from 90 degrees at CK, to 0 degrees, (the parabola has dwindled to a single point lying at B), the angle around the ground circle centered at M sweeps from 180 degrees at a launch angle of 90 degrees, (line MC) to 0 degrees at a launch angle of 0 degrees, (line MB).  When the ground line of the parabola, (line ED) lies along to the diameter of the ground circle M, the projectile has the maximum range.  This corresponds to a launch angle of 45 degrees and an angle swept through the ground circle of 90 degrees, (see picture 5).

This feels nice and intuitive, and the three mentioned points certainly match up, but if you'd like a proof of the relations between the launch angle and the angle swept out on the ground circle, see picture 6.

There's one last thing about the distance along the diameter of the ground circle, (line BC), how it's related to the angle theta in the picture above, and how that's related to the maximum height of the projectile in each parabola, but I'll save that for later.

1.  Open Access paper on the elliptical envelope of parabolic apexes.

2.  Original post on article

2.a. permalink to this post

3.  post on Da Vinci's Parabolic Compass

Tuesday, April 15, 2014

Superconductors, the London Moment, and Spin Currents

In addition to the Meissner effect pictured to the left, superconductors have other odd properties.  One of them is something known as the London moment.  It's named after the London brothers who did some of the earliest work on the properties of superconductors.  If I had to take a guess, I'd say it was specifically named after  Fritz London who authored the two volume set of books, "Superfluids"[1].  So much for the naming of the thing, here's what happens.  When a superconductor is rotated, it produces a magnetic field aligned with the axis of rotation.  The magnetic field is linearly related to the angular velocity of the superconductor by the following equation, (picture 2)[2].

where m_e is the mass of the electron, c is the speed of light, and e is the charge of an electron.

London theorized that the  magnetic field which bears his name was a result of the superconducting cooper pairs lagging behind the initial rotation of the superconductor bulk.  Thereafter, they never quite catch up and there's a relative current between the superconducting electrons and the body of the superconductor.  It's this current he reasoned that is responsible for the magnetic field.

Dr. Jorge Hirsch of UCSD has posited a different mechanism for the production of the London moment in his paper titled "Spin currents in superconductors[2]".  Hirsch theorizes that the current responsible for the London  moment is not due to the lag of superconducting electrons spinning up and trying to catch  up with the rest of the superconductor, but instead is due to counterposed currents of electrons whose spins are opposite to one another travelling around the surface of the superconductor.  The currents come in pairs.  A current of spin up electrons moving in the clockwise direction around a superconductor is always accompanied by a current of spin down electrons moving in the opposite direction.  In this way, in equilibrium, there is no detectable charge or spin currents since each pair of counter-spinning counter-traversing currents effectively cancels each other out.  According to Hirsch, these hypothetical counterposed spin currents are setup in the superconducting material when it enters the superconducting state.  A diagram of two pairs of the spin currents in a spherical superconductor is shown in picture 3[2].

Hirsch presents a very nice macroscopic explanation of how the spin currents create the London moment currents when the superconductor is rotated.  At the most basic level, the London moment current due to the spin currents compensates for the extra centripetal force on the superconducting electrons.  I'll have to leave that for another time though.

1.  London's book on superconductors

2.  Hirsch's paper on superconductors and spin currents (open access)

Sunday, April 13, 2014

Projectile Motion: Pushing the Envelope

Think everything that's publishable for say an old classical topic like projectile motions has already been published?  Turns out the old 'lob the projectile at a constant velocity in a constant gravitational field' problem is still producing.  Check out this paper from J. L. Fernandez-Chapou, A. L. Salas-Brito, and C. A. Vargas published in 2004.  It eventually made its way into the American Journal of Physics.  In the paper, the authors show that if you write down the trajectory of a projectile in terms of its launch angle and then solve for the x and y position when the projectile has reached it's maximum height, the solutions will trace out a nice little ellipse like the figure below excerpted from the arxiv version.

1.  Elliptic envelope of parabolic trajectories paper

1.a.  AJP version of the paper

Friday, April 11, 2014

Adventures in Hosing

We're trying a new type of roughing pump vacuum hosing.  We're moving away from the traditional classic, red rubber tubing and trying out Kuritech K7130 Polywire Hose!  The kids and I went on an adventure a week and a half ago to check the hosing out at Bryan Hose and Gasket!  Everyone was super nice, and it looked like if you turned up later in the day you could have free popcorn.  It was Sam's first science trip.  Jr. used to go on these all the time when were at Brookhaven National Laboratory.  The best part, besides the obvious of getting to hang out in a hose and gasket store?  On the way out, Sam and Jr. saw their first real bulldozer, (see the right hand side of the building in the picture).  After weeks of seeing them in Mater cartoons, they thought it was awesome!

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Thursday, April 10, 2014

David Hestenes of Geometric Algebra Fame to speak at Texas A&M Today

Dr. David Hestenes, (pictured to the left [2]), the original author of geometric algebra, (it was his PhD dissertation work at UCLA), will be speaking at A&M today[1].

We're learning how to do literature review matrices in our writing class, so I thought I'd try out the technique while reading Dr. Hestene's bio[2] last night.  Here are the key points I came away with

1.  Hestenes was inspired by Marcel Riesz's book "Clifford Numbers and Spinors"[3]
One day in the mathematics– engineering library I looked at a shelf of incoming new books and pulled down and some lecture notes entitled “Clifford Numbers and Spinors” by Marcel Riesz. It was about Clifford algebra as a mathematical system. I read, I think, for about 15 minutes and all of a sudden I had an epiphany. I exclaimed “Gee, differential forms and the Dirac algebra have a common algebraic structure!”…

2.  Dr. Hestenes received the Oersted medal for his work on the force concepts inventory, but delivered the accompanying lecture[4] on geometric algebra:
Yeah, except for my Oersted lecture, okay? So, I gave my Oersted lecture on elementary applications of geometric algebra instead of my educational R&D, for which the award was intended. But I related that to science education by emphasizing that what you understand about science depends critically on your facility with conceptual tools, representational tools, and mathematical tools. For example, you had to do all of your calculations with roman numerals, you wouldn’t do very well, okay?

3.  He has written a book[5] that treats advanced classical mechanics in terms of geometric algebra
I have published the first advanced book on classical mechanics worked out exclusively with geometric algebra. All equations are formulated and calculations are done without resorting to coordinates or matrices, including rotational dynamics, precessing tops, and all that. The introductory chapter is a kind of annotated history of geometric algebra

4.  The Dirac algebra is associative.  Dr. Hestenes reinterpreted the Dirac gammas in terms of vectors.
To explain since you know Dirac algebra. you know that the whole algebra is generated by the Dirac matrices, so you can understand the significance when I reinterpreted the Dirac gammas them as vectors. These vectors then generate an associative algebra, mathematically speaking, a Clifford algebra. But I developed this algebra as an encoding of geometric properties for space-time in algebraic form. I call that system space-time algebra (STA). From that viewpoint, the Pauli algebra sheds its representation by 2×2 matrices to emerge as a subalgebra of the STA. That was my second significant discovery about the Pauli algebra.

5.  Of interest to grad students with kids, (like me!).  Dr. Hestenes had four children before he was out of graduate school!
Another reason that I went to ASU and stayed there is because I was married when I was in college. I had my first child while I was in army and my second child was born on my first day in graduate school. By the time I finished my PhD I had four children. Then I went to Princeton. I have never heard of another postdoc with four children.

6.  Hestenes on spin and spinors as vectors:
The first discovery is one of the highlights of my life. And it gave me strong motivation and direction for my research. That discovery was recognition that the Pauli matrices could be reinterpreted as vectors, and their products had a geometric interpretation. I was so excited that I went and gave a little lecture about it to my father. Among other things, I said, “Look at this identity σ1 σ2 σ3 = i, which appears in all the quantum mechanics books that discuss spin. All the great quantum physicists, Pauli, Schroedinger, Heisenberg and even Dirac as well as mathematicians Weyl and von Neumann, failed to recognize its geometric meaning and the fact that it has nothings to do with spin. When you see the Pauli sigmas as vectors, then you can see the identity as expressing the simple geometric fact that three orthogonal vectors determine a unit volume. Thus there is geometric reason for the Pauli algebra, and it has nothing whatsoever to do with a spin.

7.  Dr. Hestenes thinks there are opportunities for further research in superconductivity
T: Because it just occurs to me that we need a somewhat new theory of electron probably to solve the questions of superconductivity. Do you feel a need of that sort?
H: I feel more than a need. I think I know at least one way that the theory should be changed. The standard theory of superconductivity is not as successful
as people make it out to be. When you get near the critical point they have a renormalization theory to explain what happens. But renormalization theory doesn’t get the correct result for the critical point. And not just at the critical point! The deviations of theory from the experimental data increase as you get closer
and closer to the critical point. So, what is going on there? Here is my hypothesis: the electrons have this internal Zitter motion, and as you approach the critical point there is an increase in Zitter correlations, that is, in resonances between Zitter motions of different electrons. As temperatures increase correlations are destroyed by thermal fluctuations. I submit this as a general explanation for all critical phenomena in condensed matter systems.
With a nod to the Smitten Kitchen for the new post style

Picture of the Day:
A vacant factory in Savannah, GA

1.  Seminar announcement

2.  Taşar, M., Bilici, S., Fettahlıoğlu, P., "An Interview with David Hestenes: His life and achievements", Eurasia Journal of Mathematics, Science & Technology Education, 8(2), (2012), 139-153

3. Marcel Riesz's book "Clifford Numbers and Spinors", available open access:

4.  Dr. Hestenes Oersted Lecture

5.  Hestenes book on classical mechanics