Today is the first day of the last week of class, hallelujah. Unfortunately, it's also the first class on rotational motion and angular momentum. This is unfortunate because it's the hardest material in the course-- angular momentum doesn't behave in as intuitive a manner as linear momentum, and the math involved is the most complicated of anything we do in the course.
This mostly has to do with the vector product, or "cross product." Angular momentum can be written as the product between the position vector from the axis of rotation to the moving object and the linear momentum of the moving object. The resulting vector is at right angles to both of the original vectors.
This is just plain weird, but it leads to most of the interesting properties of angular momentum. Unfortunately, it also requires a good deal of careful attention to detail to get it right. And we've got two classes in which to discuss it, two classes that fall in the first week of June, when everybody's brains are completely shot (faculty included), and all thoughts are on the imminent summer.
It's a shame, because this is the coolest material in intro mechanics. But it's almost impossible to do it justice in these conditions. As a result, I end up kind of dreading the whole thing.
Stupid academic calendar.
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The vector cross product (a special case of tensor bling) is chiral and odd-parity, A(X)B = -B(X)A. Conservation of angular momentum the observable is coupled to the symmetry of vacuum isotropy through Noether's theorems. Noether demands continuous or at least so approximatable (Taylor series) symmetries. Parity is an external symmetry (couples to translation and rotation) that is absolutely discontinuous. Conservation of angular momentum can suffer a parity violation without paradox or contradiction.
Covariance with respect to reflection in space and time is not required by the Poincaré group of Special Relativity or the Einstein group of General Relativity. Given two lumps comprised of chemically identical, opposite geometric parity atomci mass distributions... all bets are off. Quantum gravitation theories supplement Einstein-Hilbert action with an odd-parity Chern-Simons term, f(x) = -f(-x), arxiv:0811.0181.
Run a parity Eötvös experiment. Quartz single crystals in enantiomorphic space groups P3(1)21 versus P3(2)21 qualify for symmetry, with dense atom packing of 0.01256 nm^3/atom. Eur. J. Mineralogy 2, 63 (1990), J. Solid State Chem. 36, 371 (1981). Glycine gamma-polymorph has lighter atoms more densely packed, enantiomorphic space groups P3(1) versus P3(2), 0.007869 nm^3/atom. Acta Crystallogr. B36, 115 (1980).
Somebody should look.
Set up the Frenet Frame for the center of mass of the object/particle, moving on the path gamma. The angular momentum vector is the Darboux vector given by the sum of the torsion scalar times the T-basis vector in the Frenet frame plus the curvature scalar times the B-basis vector in the Frenet frame. It all drops out obviously and beautifully - all of its symmetries fall out in terms of the derivative vectors of the Frenet frame basis vectors. e.g. (Darboux vector) cross product T = T prime, which you have already calculated.
You aren't supposed to admit that! The faculty are super-human!
I think Uncle Al was trying to tell you and everyone else that the vector cross product is anti-commutative, but mixed up that detail with its unusual behavior under a parity inversion.
Side note:
I always make a big point about it being a non-Abelian (non commutative) operator. Students routinely mess up vxB calculations in E+M because of that property.
The result of the vector cross product is a pseudo- or axial-vector, not a vector. A parity operation flips the sign of a vector, but the sign of a pseudo-vector remains unchanged (because the signs of both of the vectors used to construct it change).
The "V-A" (a particular combination of vector and axial vector) weak interaction responsible for beta decay is needed so those interactions will violate parity. This last bit is probably what Uncle Al thought he was talking about based on the rest of his post.
riddle: what is purple and doesn't commute?
ans: an non-abelian grape.
heh.
Hi Chad,
You're using M&I, right? I don't know your syllabus, but maybe next semester you could skip Chapter 7 and put it at the end after the quantized angular momentum in Chapter 10. The quantum stuff in Ch 7 is presented in M&I in a fairly straight forward manner and is a nice review/application of previous topics...which could be good when everyone's brain is mush.
Georgia Tech uses this approach:
http://www.physics.gatech.edu/academics/Classes/spring2009/2211/K_to_N/…
I agree: angular momentum is the fun stuff!
Good luck!
The use of the Gibbs vector product in physics always annoyed me, and it took until after I graduated to find out why: It's almost always a geometric outer product in disguise.
Angular momentum isn't measured by a directed line segment, but a directed plane segment. A bivector just happens to be isomorphic to a vector in 3-space, hence the confusion.
The use of the Gibbs vector product in physics always annoyed me, and it took until after I graduated to find out why: It's almost always a geometric outer product in disguise.
Angular momentum isn't measured by a directed line segment, but a directed plane segment. A bivector just happens to be isomorphic to a vector in 3-space, hence the confusion.
Pseudonym mentions something in passing that you might use to inform your teaching, Chad. If, as is common at my CC, students taking E+M are also taking calc III, you should know about the tangent, normal, and binormal vectors that show up early in that course. (They correspond to the velocity, centripetal force, and helicity of a particle moving on the thread of a screw.) It is also true that the angular momentum and torque (moment to the engineering crowd) is much like the normal vector that defines the area of an oriented plane, the same oriented area you use in Gauss' Law. Making these connections helps learning in both physics and math.
My nit pick is that an outer product results in a tensor, a 3x3 matrix in the case of the outer product of two 3 vectors. Although the axial vector resulting from the cross product is, technically, a tensor - it is not THAT tensor. I usually mention that also, since they rarely hear about it in a math class.
CCPhysicist is necessary but not complete. A spinning sphere has helicity but not chirality (viewed from the opposite spin axis pole it is reversed). Relativistically translating helicity is chiral. Beta-rays slightly decomposing a solid amino acid racemate enrich the remainder with one enantiomer at the surface few microns when they are still energetic, possibly sourcing biological homochirality (e.g., Bonner).
All scalene triangles are chiral in 2-D (orientable surface). In an arbitrary number of dimensions geometric parity divergence of a set is calculable on a scale of CHI = 0(achiral) to 1(perfectly parity divergent) inclusive,
CHI = (d)[Min{P,R,t}D^2]/4T
CHI is globally minimum for all rotations (R) and translations (t) for all pairwise correspondences or graph automorphisms (P) of the set where d is the Euclidean dimension, D^2 is the sum of the N squared-distances between the set and its parity inversion for a fixed pairwise correspondence with coincident centers of mass, and T is the geometric inertia of the set (Petitijean). Angular momentum scales geometric chirality.
Fundamental physical theory arises from fundamental symmetries, S((U2)xU(3)) or U(1)xSU(2)xSU(3) for the Standard model. Theory chokes on geometric parity divergences, ad hoc inserting it as empirically necessary. No observation prevents gravitation from displaying parity asymmetry. A parity Eotvos experiment opposing enantiomorphic quartz is the test. The first quantitative Equivalence Principle tests ran in the late 1500s. It would not be horrible if more than 400 years of failures added one more failure - or the first success. Cowards.
(Metric gravitation in pseudo-Riemannian spacetime would fall to teleparallel gravitation in Weitzenböck spacetime. GR is then a heuristic accurate except for the special case - that includes spin-orbit coupling in extreme relativistic binary stellar systems.)
Trimesters/Quarters or whatever you want to call them do have that serious downside. On the other hand just think about how positively leisurely semesters will seem to any of your students who go to graduate school on that system. Those that continue on the same calender will be prepared for the somewhat fast pace.
Uncle Al, a little brain-picking if you would. Going from the macro scale to the micro scale, at what exact point does the concept of spin become a quantized abstraction that no longer necessarily represents anything we can mentally picture?
On the macro scale, pretend you are on one of the very out-most stars of the Milky Way. You are moving a lot faster than GR says you should be. You look inward at the galactic center and notice compression waves making inward stars appear to be in spiral arms because they are "bunching up" or "thinning out" as the wave passes through them (moving much faster than they can move.) This bunching and thinning would involve accelerations and decelerations of the stars:
1) What is the energy source of the compression wave and is it on-going or a one-time vestigial impulse left over from galaxy formation?
2) Would an observer on one of the inward stars being accelerated FEEL anything, or is it a free fall situation all the way?
3) A different observer is on a spaceship orbiting a black hole in a trajectory so highly elliptical that she is near the speed of light when swooping near the B.H. and slowing to 55mph near the center point of the ellipse. Isn't it a bit strange she feels no acceleration at all at any time?
4) The B.H. was at one foci of the ellipse, but as she continues to travel outward she accelerates to near-C again as she rounds the outward foci. So, is there any possibility that outward foci is the center of mass of all dark matter in the system? Do you believe in Dark Matter?
On the macro scale, pretend you are on one of the very out-most stars of the Milky Way. You are moving a lot faster than GR says you should be. You look inward at the galactic center and notice compression waves making inward stars appear to be in spiral arms because they are "bunching up" or "thinning out" as the wave passes through them (moving much faster than they can move.) This bunching and thinning would involve accelerations and decelerations of the stars:
1) What is the energy source of the compression wave and is it on-going or a one-time vestigial impulse left over from galaxy formation?
2) Would an observer on one of the inward stars being accelerated FEEL anything, or is it a free fall situation all the way?
3) A different observer is on a spaceship orbiting a black hole in a trajectory so highly elliptical that she is near the speed of light when swooping near the B.H. and slowing to 55mph near the center point of the ellipse. Isn't it a bit strange she feels no acceleration at all at any time?
4) The B.H. was at one foci of the ellipse, but as she continues to travel outward she accelerates to near-C again as she rounds the outward foci. Could the outward foci be the center of mass of all dark matter in the system?
What I really meant to ask with all that double-posted blather is that if the astronaut in the spaceship zooming near the B.H. in a highly elliptical trajectory feels no acceleration along the geodesic due to relativity principles, why would she feel the tidal forces? Is she maybe not feeling the tidal force, but it is the relativistic stretching out of herself and her vessel at right angle to her direction of travel due to the near-c velocity that will appear as a tidal effect?
What blows my mind is that Vector Products, properly speaking, happen ONLY in the 3rd dimension and the 7th dimension. We teach our students about the 3 dimensions of space, and how they connect to the 4th dimension of time. But are there beings in 7-D who consider our cosmos a trivial special case?
Seven-dimensional cross product
From Wikipedia, the free encyclopedia
In mathematics, the seven-dimensional cross product is a binary operation on vectors in a seven-dimensional Euclidean space. It is a generalization of the ordinary three-dimensional cross product. The seven-dimensional cross product has the same relationship to the octonions as the three-dimensional cross product does to the quaternions.
Nontrivial binary cross products exist only in 3 and 7 dimensions. There are no higher-dimensional analogs....