- #36
Set of possible rotations of a 3D object and the rotation history
In summary, the conversation discusses the concept of "rotation space" and how it relates to representing the rotational history of an object. The space is described as a solid ball with opposite points on the surface identified and each point representing a direction and angle less than or equal to π. A path in this space represents a rotation about a specific axis and angle. The conversation also touches on the idea of precession and how it relates to paths in rotation space. The conversation ends with a discussion on the identification mapping of the 3 ball as the topological cone on the 2 sphere and an example of a path of rotations through an angle of π/2.
- #37
bolbteppa
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If the flag is to not be completely superfluous it should at the very least capture that a ##2 \pi## rotation brings in the minus sign, however (page 6):
"The angle doubling leads to the curious feature that when ##\theta= 2 \pi## (a single full rotation) the spin rotation matrices all give ##-I##. It is not that the flagpole reverses direction — it does not, and neither does the flag — but rather, the spinor picks up an overall sign that has no ready representation in the flagpole picture."
My reading of MTW (page 1157 on from your last post) is again that they do not give you spinors from this idea of flags, they try to describe the already-established notion of spinors using flags, and earlier (very badly) motivate spinors by arguments that amount to the arguments of the PDF of the OP.
"The angle doubling leads to the curious feature that when ##\theta= 2 \pi## (a single full rotation) the spin rotation matrices all give ##-I##. It is not that the flagpole reverses direction — it does not, and neither does the flag — but rather, the spinor picks up an overall sign that has no ready representation in the flagpole picture."
My reading of MTW (page 1157 on from your last post) is again that they do not give you spinors from this idea of flags, they try to describe the already-established notion of spinors using flags, and earlier (very badly) motivate spinors by arguments that amount to the arguments of the PDF of the OP.
- #38
Spinnor
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All right then, into the trashcan of science with the flagpole and flag. I wish there was something just as intuitive to replace it.
Thank you.
Thank you.
- #39
bolbteppa
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I would be shocked if one could get more intuitive than the notes of the OP for this stuff 
This perspective of spinors from the POV of connectivity of Lie groups is very important for understanding why (finite dimensional) spinors exist for the subgroup SO(n) of GL(n) but not GL(n) itself
https://www.physicsforums.com/threads/why-there-are-no-spinors-for-gl-n.240240/
A nice simple proof of this would be welcome (hint hint readers)
This perspective of spinors from the POV of connectivity of Lie groups is very important for understanding why (finite dimensional) spinors exist for the subgroup SO(n) of GL(n) but not GL(n) itself
https://www.physicsforums.com/threads/why-there-are-no-spinors-for-gl-n.240240/
A nice simple proof of this would be welcome (hint hint readers)
- #40
lavinia
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I thought it would be helpful to elaborate on the parameterization of ##SO(3)## given in post #1. This is the 3 dimensional ball ##B^3## with antipodal points on its boundary identified.
Topologically ##B^3## modulo these identifications is homeomorphic to the real projective 3 space ##RP^3##. The book is showing that ##SO(3)## viewed as a topological space is homeomorphic to ##RP^3##.
In topology, the real ##n## dimensional projective space ##RP^{n}## is the quotient space of the ##n## dimensional unit sphere in ##R^{n+1}## with its antipodal points identified. Antipodal points come in pairs and are the intersections of lines through the origin with the unit ##n## sphere. They may be thought of as opposite geographic poles. The quotient mapping ##S^{n}→RP^{n}## that identifies antipodal points is 2 to 1, is continuous and any small enough ball in ##S^{n}## - a ball that contains no antipodal points e.g. a polar ice cap - is mapped homomorphically into a small ball in ##RP^{n}##.This follows from the definition of the quotient topology and shows that ##RP^{n}## is a closed ##n## dimensional manifold and that ##S^{n}## is a 2 fold covering space of it.Consider now the northern hemisphere of ##S^{n}## together with the equator - which is the ##n-1## dimensional sphere ##S^{n-1}##. This space is an ##n## dimensional topological ball as can be seen by projecting it vertically onto the ##n## dimensional plane containing the equator. Under the quotient map ##S^{n}→RP^{n}## antipodal points on the equatorial ##n-1## sphere are identified and in the northern hemisphere proper, each point is mapped to a unique point. Every point in ##RP^{n}## is represented either by a unique point if the northern hemisphere or by a pair of antipodal points on the equator. In the case of ##S^3## the northern hemisphere is a three dimensional ball So its image in ##RP^3## is a three dimensional ball with antipodal points on its boundary 2-sphere identified. This is exactly the same topological description and in post #1 and shows why ##SO(3)## is topologically the same as the real projective space ##RP^3##. So ##SO(3)## is more than just a group of matrices. It is a closed three dimensional manifold.
The only difference between this description and the description in post #1 is the interpretation of parameters. Instead of axes of rotation, one has directions along great circles emanating from the north pole. Instead of an angle of rotation, one has the distance from the north pole. The angle ##θ## lies between ##0## and ##π/2## rather than between ##0## and ##π##. Points along the equator all have an angle of ##π/2##. The curve illustrated in post #1, the straight line through the center of the ball with end points on the boundary, corresponds now to a half great circle through the north pole. Its end points are antipodal so it projects to a closed loop ##γ## in projective space.
Topologically ##B^3## modulo these identifications is homeomorphic to the real projective 3 space ##RP^3##. The book is showing that ##SO(3)## viewed as a topological space is homeomorphic to ##RP^3##.
In topology, the real ##n## dimensional projective space ##RP^{n}## is the quotient space of the ##n## dimensional unit sphere in ##R^{n+1}## with its antipodal points identified. Antipodal points come in pairs and are the intersections of lines through the origin with the unit ##n## sphere. They may be thought of as opposite geographic poles. The quotient mapping ##S^{n}→RP^{n}## that identifies antipodal points is 2 to 1, is continuous and any small enough ball in ##S^{n}## - a ball that contains no antipodal points e.g. a polar ice cap - is mapped homomorphically into a small ball in ##RP^{n}##.This follows from the definition of the quotient topology and shows that ##RP^{n}## is a closed ##n## dimensional manifold and that ##S^{n}## is a 2 fold covering space of it.Consider now the northern hemisphere of ##S^{n}## together with the equator - which is the ##n-1## dimensional sphere ##S^{n-1}##. This space is an ##n## dimensional topological ball as can be seen by projecting it vertically onto the ##n## dimensional plane containing the equator. Under the quotient map ##S^{n}→RP^{n}## antipodal points on the equatorial ##n-1## sphere are identified and in the northern hemisphere proper, each point is mapped to a unique point. Every point in ##RP^{n}## is represented either by a unique point if the northern hemisphere or by a pair of antipodal points on the equator. In the case of ##S^3## the northern hemisphere is a three dimensional ball So its image in ##RP^3## is a three dimensional ball with antipodal points on its boundary 2-sphere identified. This is exactly the same topological description and in post #1 and shows why ##SO(3)## is topologically the same as the real projective space ##RP^3##. So ##SO(3)## is more than just a group of matrices. It is a closed three dimensional manifold.
The only difference between this description and the description in post #1 is the interpretation of parameters. Instead of axes of rotation, one has directions along great circles emanating from the north pole. Instead of an angle of rotation, one has the distance from the north pole. The angle ##θ## lies between ##0## and ##π/2## rather than between ##0## and ##π##. Points along the equator all have an angle of ##π/2##. The curve illustrated in post #1, the straight line through the center of the ball with end points on the boundary, corresponds now to a half great circle through the north pole. Its end points are antipodal so it projects to a closed loop ##γ## in projective space.
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- #41
lavinia
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Closed loops in projective space
As was shown in post #40 projective space ##RP^{n}## is the quotient of the ##n##-sphere ##S^{n}## with antipodal points identified. A loop in ##RP^{n}## lifts to two antipodal paths on the sphere and beginning and ending points on one path are antipodal to beginning and end points on the other. If the paths are also closed loops the lift is two antipodal closed loops on the sphere. If the paths are not loops then the end points of each are antipodal. In this second case, the two paths fit together to make a single loop.
So every closed loop in projective space is covered twice by its preimage on the sphere. The preimage is either two antipodal loops or two antipodal paths that join together at their antipodal ends to form a single loop. In this second case, the closed loop does not project back onto the original loop but instead projects to its double, the loop that wraps around it twice. This is because each piece wraps around it once.
Each of these loops is contractible - in fact every closed loop on a sphere of dimension greater than 1 is contractible. - and any contraction projects to a contraction in projective space. A contraction of either of the antipodal closed loops projects back to a contraction of the original loop that they come from, while a contraction of the. spliced together loop projects to a contraction of the double of the original loop.
So every closed loop in a projective space of dimension greater than one is either contractible or its double is contractible. The double of the diameter line in post #1 is contractible but since it is not closed in the 3 ball, it may not be contractible by itself.
The idea of the proof that every closed loop on the sphere is contractible.
If a closed loop on the sphere misses at least one point, then it can be contracted along great circles through one of the missing points. If the loop is space filling, then it can be first continuously deformed into a loop that is not, then contracted. Proving that a space filling loop can be deformed to a non-space filling loop requires a little work and is the only hard part of the proof.
Notes:
- A space filling curve is a continuous path that completely covers a region of space. Every point in the region - for instance of a square or of a cube. - is crossed by the path. Such paths can be shown to exist as the uniform limits of certain sequences of continuous paths.
- The formal definition of a contraction of a loop
One imagines a contraction of a loop as a stretched rubber band that shrinks as it releases tension. At each point in time, the band forms a smaller loop until finally it has zero tension. Formally this is a continuous 1 parameter family of loops and may be described as a continuous map from ##H:S^1×[0,1]→X## from a circle Cartesian product an interval into a topological space ##X## which at time zero is the starting loop and at time one is the constant loop. The map ##H## is called a homotopy and is similar to a variation except that it is only required to be continuous rather than smoothly differentiable and end points are allowed to wander. The idea of a homotopy is not restricted to ##S^1## and makes sense for any topological space.
As was shown in post #40 projective space ##RP^{n}## is the quotient of the ##n##-sphere ##S^{n}## with antipodal points identified. A loop in ##RP^{n}## lifts to two antipodal paths on the sphere and beginning and ending points on one path are antipodal to beginning and end points on the other. If the paths are also closed loops the lift is two antipodal closed loops on the sphere. If the paths are not loops then the end points of each are antipodal. In this second case, the two paths fit together to make a single loop.
So every closed loop in projective space is covered twice by its preimage on the sphere. The preimage is either two antipodal loops or two antipodal paths that join together at their antipodal ends to form a single loop. In this second case, the closed loop does not project back onto the original loop but instead projects to its double, the loop that wraps around it twice. This is because each piece wraps around it once.
Each of these loops is contractible - in fact every closed loop on a sphere of dimension greater than 1 is contractible. - and any contraction projects to a contraction in projective space. A contraction of either of the antipodal closed loops projects back to a contraction of the original loop that they come from, while a contraction of the. spliced together loop projects to a contraction of the double of the original loop.
So every closed loop in a projective space of dimension greater than one is either contractible or its double is contractible. The double of the diameter line in post #1 is contractible but since it is not closed in the 3 ball, it may not be contractible by itself.
The idea of the proof that every closed loop on the sphere is contractible.
If a closed loop on the sphere misses at least one point, then it can be contracted along great circles through one of the missing points. If the loop is space filling, then it can be first continuously deformed into a loop that is not, then contracted. Proving that a space filling loop can be deformed to a non-space filling loop requires a little work and is the only hard part of the proof.
Notes:
- A space filling curve is a continuous path that completely covers a region of space. Every point in the region - for instance of a square or of a cube. - is crossed by the path. Such paths can be shown to exist as the uniform limits of certain sequences of continuous paths.
- The formal definition of a contraction of a loop
One imagines a contraction of a loop as a stretched rubber band that shrinks as it releases tension. At each point in time, the band forms a smaller loop until finally it has zero tension. Formally this is a continuous 1 parameter family of loops and may be described as a continuous map from ##H:S^1×[0,1]→X## from a circle Cartesian product an interval into a topological space ##X## which at time zero is the starting loop and at time one is the constant loop. The map ##H## is called a homotopy and is similar to a variation except that it is only required to be continuous rather than smoothly differentiable and end points are allowed to wander. The idea of a homotopy is not restricted to ##S^1## and makes sense for any topological space.
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