Multivector momentum

[update 2014-12-10: This post is definitely not quite right. I'm trying to understand units and torsors better. Also, how then to fit heat/temperature better in with the others. Hopefully I'll write about these things one day, but fixing this post seems likely to remain beyond me.]

I trained as a Mathematician, and I try at times to understand some developments in modern Mathematics. Mostly I do this by following John Baez who writes beautifully so that you imagine that you understand it. I’m only vaguely following his current interest in networks, but I loved this diagram (from http://johncarlosbaez.wordpress.com/2013/04/23/network-theory-part-29/) for an unrelated reason:

	displacement:	flow:	momentum:	effort:
Mechanics: translation	position	velocity	momentum	force
Mechanics: rotation	angle	angular velocity	angular momentum	torque
Electronics	charge	current	flux linkage	voltage
Hydraulics	volume	flow	pressure momentum	pressure
Thermal Physics	entropy	entropy flow	temperature momentum	temperature
Chemistry	moles	molar flow	chemical momentum	chemical potential

The first row (after the headings) is the familiar fact that a moving body has momentum which keeps it moving in a straight line at constant speed. To change that requires effort: the application of a force. We are equally familiar in life with the fact that a spinning body (such as a top, or the Earth) has angular momentum which keeps it spinning, and you have to put in effort to stop it.

In the 4th line Dr Baez is thinking of liquids in pipes, but I am more interested in unconstrained gases. If you imagine an explosion then that has pressure momentum which will keep it expanding forever. It requires effort to stop the expansion and that effort would be pressure.

I also want another row which might be the same as Dr Baez’s 6th line. At any rate the momentum in this case is heat. Things will keep the same amount of heat unless some effort (heating/cooling) is applied to change it. This might seem different to the others, since the others are about movement, but actually heat is just internal movements within the matter. The temperature of a gas is proportional to the kinetic energy per molecule.

Now I need to take a detour before I can put these together.

Multivectors

A vector is something with magnitude and direction. Speed is just a number, but velocity (and momentum and force) are vectors because they also include the direction. We imagine vectors as arrows in space with the length representing the magnitude of the vector. You can slide them around: they don’t start at a particular place. Our vectors are three dimensional: every vector can be made of a bit of x, a bit of y and a bit of z.

You can combine two vectors to make a bivector. This is called the exterior product. To imagine this we place the 2nd vector so that it starts where the first finishes. This makes a little parallelogram floating in space, and the area is the magnitude of the bivector. However the bivector is not really a parallelogram: it can be any shape in that plane with that area. And there is a natural way of adding bivectors and they are also 3 dimensional: every bivector can be made of a bit of xy, a bit of yz and a bit of zx.

The exterior product of 3 vectors is a trivector. We can visualize this as a bit of volume with the 3 vectors at edges, but once again it is just the amount of volume that counts not the shape. Trivectors are just one dimensional, since each is just a multiple of the 1x1x1 xyz volume element.

There are nice diagrams of all these at Wikipedia’s Geometric Algebra page: http://en.wikipedia.org/wiki/Geometric_algebra.

It turns out to often be a useful idea to combine our 3 dimensions of vectors, 3 of bivectors, 1 of trivector plus 1 more for scalars (scalars are just numbers). This makes an 8 dimensional space of multivectors. These are mostly particularly useful when combined with a way to multiply them, but we won’t get to that here.

Momentum types

Ordinary momentum is a vector. It has magnitude and direction.

Angular momentum is best seen as a bivector. The area of the bivector is the magnitude, and its orientation (perpendicular to the axis of rotation) determines the rotational geometry.

Since pressure acts in all directions at once, it is natural to see it as a trivector.

Finally we have heat which is intrinsic to the matter and not involved in directions. It is just a scalar value.

It is easy to combine these together into a single multivector value. The question is how meaningful that is.

One aspect is this: Are there other mechanical momentums that this leaves out? I conjecture that for an infinitesimal amount of matter this is all there is.

Speaking of infinitesimal amounts of matter: It is tempting to think that a small amount of matter can’t have much angular momentum. But actually the smaller matter is, the faster it is able to rotate. Even electrons have significant spin (though the mechanicalness of that might be in some doubt). I wonder if the study of fluid mechanics takes adequate account of small rapidly spinning vortices?

Implications

It seems cute, but that is not a justification. It does suggest two lines of investigation:

One is to study fluid mechanics in full generality starting with infinitesimal amounts of matter having these 4 types of momentum. The hope will be that some of the research on Clifford Analysis will turn out to be useful.

The other is to use this in computer simulations of oceans, atmosphere or other fluid situations. These are typically done by dividing the matter to be simulated into little cuboids. The cuboids have some physical characteristics, and the value at the next step is determined from the current value plus the values in neighboring cuboids (plus other forcings, i.e. effort, that may be specified). The hypothesis is that this 8 dimensional value, across 4 types of momentum, is the optimal choice for the value to be stored in each cuboid.

Saving Test Cricket

It is nice to have the 3 levels of Cricket, corresponding roughly to sprint, middle distance and marathon events in Athletics. However Test Cricket seems to mostly survive on cultural links to the past. I don't have a problem with the duration of the game or the shortage of action, but there are problems which need to be, and can be, fixed:

• Timeless Tests had serious practical problems that were exposed in the last years of the 1930s. The solution of limiting Tests to 5 days and allowing draws has been worse (though it has produced the occasional exciting moment).
• Games need a mercy rule so that they don't go longer than necessary when they get one-sided. Teams batting on and on when they already have more than enough runs is bad for players and spectators.
• The deterioration of wickets through the course of a match gives an excessive advantage to the team winning the toss.

Luckily I have a simple solution to all 3 problems:

The solution is to have the two teams' innings in parallel (as near as practicable). Every 30 overs the teams switch over who is batting. Except that, when one team is behind on runs and has lost more wickets then they are given 2 consecutive batting segments. Here's why this solves all the problems above:

• The deterioration of the wicket is no longer a problem because it affects both sides equally. In fact it is a good thing because:
• Wickets can be made to start well but not last longer than 4 days so that wickets fall rapidly from the 5th day onwards. This stops matches going too long, but avoids the need for a time limit. It also encourages teams to score quickly while the wicket is good.
• This is a perfect mercy rule. The winning team would rarely do more than 30 overs batting beyond what is needed to win the match.

[30 overs is chosen to fit with the natural breaks in the game.]

group selection of humans

Our species is social, and we have a lot of adaptations designed to support cooperation. The obvious explanation is group selection: groups with those adaptations were more successful than those without. The trouble is that it is very hard for group selection to succeed. Cheating genes, which take advantage of the cooperation of others without reciprocating, seem certain to overwhelm the honest players.

This confused me for a long time, until I read Jared Diamond's "The World Until Yesterday". The key point is the large amount of inbreeding within primitive villages. Most people marry cousins, so that all the people in the village are very closely related. There is some gene transfer with neighbouring villages, but little beyond that.

Group inbreeding potentially creates a situation similar to social insects, where every individual is closely related to everyone else, and in particular to the reproducing females. This allows the group to function as the unit of evolution so that group selection can operate and cooperation evolves. So it seems obvious that this must be the normal (i.e. pre-civilization) human situation.

Of course inbreeding can be taken too far, and we see that it is natural for high status individuals to have the privilege of partnering outside the group. So how much inbreeding is there? Let me guess that a balance is maintained. Groups with too much inbreeding lose from the direct genetic cost. Group with too little inbreeding lose by failing to maintain group selection and being invaded by cheating genes.

This is non-expert speculation. However it is an important area to understand because it has obvious implications for the future of humanity. We have left our traditional lifestyle behind so quickly that the evolutionary effects have not had time to reveal themselves.

[update: "can't"=>"can". A senior moment.]

[update: It isn't that some groups have cooperation and some don't. What evolves to support groups is an instinct to suppress non-cooperation within the group. We hate freeloaders and cheats. But we need groups to have common genes from inbreeding for us to get value out of those "suppression of non-cooperation" genes. Otherwise we gain by avoiding the costs of that suppression effort. This becomes significant when we move to culture-based groups: see What is Culture for?]