Wherever you are, whatever you do, there is a certain special direction that takes you out of the infinite labyrinth, and leads to the reincarnation cycle getting closer and closer to the ideal path. Ordinary people do not know about it. Warriors do know.
Beyond a certain point there is no return. This point has to be reached. Franz Kafka
I am discussing geodesic lines on the three-sphere, geodesics of the left-invariant metric determined by asymmetric rigid body. There is a special, very special class of geodesics there. At every point there is a special direction. If you start the geodesic in this special direction – it has, at both ends, in the future and in the past – a limit cycle/circle. Enough esoteric talk. Let’s go to the math. The following math describes one such geodesic line. Other are obtained by left translations.
are moments of inertia of our rigid body, ordered as
The story is this. The body can rotate uniformly about its middle axis forever. Either left or right. These are two circles in the rotation group that, using its double cover, topologically is Stereographic projection maps circles into circles. So they become two circles in These circles:
But there is another possibility, when this uniform rotation along the middle axis is only asymptotic. It happens in the infinite past and in the infinite future. But only approximately with the real life, and excluding the short metamorphosis period. This is the trajectory described by the formulas above. Here is the trajectory for time t from to :
And here is this trajectory together with asymptotic circles. Here the red circle is for the bad past, the blue circle, for the good future. Under microscope it reveals rich structure – infinite mystery.
Saturday. It was warm and sunny. Went for a bicycle raid. Working on generating the exceptional geodesic for d=0.5.
But, until that is ready, I have made another geodesic line on the three-sphere, this time for d=0.4999999 and 180 degrees rotation in 4D.
Update: I have produced the geodesic for d=1/2. Here is the result for t going from -2000 to 2000:
It is a great surprise for me. Is it really one closed line? Or, under microscope, it will split into many-many lines, very close, but with a subtle structure?
Update at 22:12
Progress. Here is one point on the trajectory (I created animation to see how it is moving)
And here is zoom on the line when this point was:
The line is not a line. These are many many lines, but very close to each other.
I can’t help it. I am obsessed with geodesics.
Quoting from Encyclopedia of Mathematics:
The notion of a geodesic line (also: geodesic) is a geometric concept which is a generalization of the concept of a straight line (or a segment of a straight line) in Euclidean geometry to spaces of a more general type. The definitions of geodesic lines in various spaces depend on the particular structure (metric, line element, linear connection) on which the geometry of the particular space is based. In the geometry of spaces in which the metric is considered to be specified in advance, geodesic lines are defined as locally shortest. In spaces with a connection a geodesic line is defined as a curve for which the tangent vector field is parallel along this curve. In Riemannian and Finsler geometries, where the line element is given in advance (in other words, a metric in the tangent space at each point of the considered manifold is given), while the lengths of lines are obtained by subsequent integration, geodesic lines are defined as extremals of the length functional.
Geodesic lines were first studied by J. Bernoulli and L. Euler, who attempted to find the shortest lines on regular surfaces in Euclidean space.
The play takes place at Mr. Jourdain’s house in Paris. Jourdain is a middle-aged “bourgeois” whose father grew rich as a cloth merchant. The foolish Jourdain now has one aim in life, which is to rise above this middle-class background and be accepted as an aristocrat. To this end, he orders splendid new clothes and is very happy when the tailor’s boy mockingly addresses him as “my Lord”. He applies himself to learning the gentlemanly arts of fencing, dancing, music and philosophy, despite his age; in doing so he continually manages to make a fool of himself, to the disgust of his hired teachers. His philosophy lesson becomes a basic lesson on language in which he is surprised and delighted to learn that he has been speaking prose all his life without knowing it.
We are speaking geodesics all the time without knowing it.
On the regular two-dimensional sphere geodesics are parts of great circles
A smart ant would travel from A to B along path c, from A to C along b, and from B to C along a. These are lines as straight as only possible, that are on the surface. Geodesics on the ideal sphere are all of finite length, the maximal ones are closed, they are great circles.
But if the sphere is not so ideal, if it is a three-axial ellipsoid, even if a little bit deformed from being ideally spherical, then everything changes. Geodesics do not close, they can behave in a strange way. Again, on a small scale, they are as straight as possible, but going always “straight” we can, after long enough time, find ourselves almost everywhere. This is a difficult subject – see Geodesics on an ellipsoid.
We are not on an ellipsoid, but we are on something similar in spirit: we are on three-dimensional sphere endowed with three-axial geometry. It is the three moments of inertia that determine geometry of our After all is the whole universe of the rigid body that is rotating about its center of mass. Three parameters determine rotation matrix. Unit quaternions which we use do the same. But not all rotations are equal. Some are for the body “harder” than other, some are “easier”. This depends whether the instantaneous rotation axis is close to the one with the greatest moments of inertia, or the smallest one. Therefore also “distance” between two rotations that somehow measures “effort” exerted by the body, will depend on the direction in
I know, we will have to learn all that. But, on the other hand, it is something that should be expected. For some reason that we do not fully understand, lot fundamental equations of dynamics can be derived from “least action principle”. But least action can be translated into “shortest way”- with the appropriate definition of the distance. Therefore, perhaps, all physics can be converted into geometry, and the Nature is simply choosing always the shortest path (or, sometimes, the longest).
So, here is again the same animation, but there are more details.
You will have to click on the image to open animation in a new window. There are 91 frames. I am rotating the observer every one degree from 0 to 90 degrees. Something rather strange happens around 84 degrees. That is the extracted frame above. The animation represents one geodesic line. I mean part of it. Time goes from -2000 to 2000 (of whatever units we use). It can go from minus infinity to infinity. I have no idea how the picture would change with the extension of the length of time. Probably it would be getting denser and denser. The phenomenon here may be similar to the one exhibited by the famous chaotic Lorentz attractor.