Topology of the set of possible orbits

[Linguofreak]

I've been mulling lately over what the set of orbits around a gravitating body looks like as a topological space. I'm not a math or physics major of any kind, so my grasp on topology is quite weak, but I think I've managed to identify a few things (correct me if I've misidentified things, I've done this by visualization and what I've read about the spaces in question, not by doing rigorous math. Also, given my mathematical nonexpertise, I may misuse terminology even if I have the right idea. I apologize in advance.):

1) The space of orbital planes (ignoring eccentricity, semi-major axis, etc.) around a body without specifying orbital direction is the real projective plane, since we can identify each plane with an axis, so our space of orbital planes reduces to the space of lines through the origin of 3D euclidean space.

2) The space of orbital planes around a body with orbital direction specified is the sphere.

3) If we extend (1) by adding orbital phase as a coordinate, we get SO(3).

4) If we extend (2) by adding orbital phase as a coordinate, we get SU(2). Have I gotten this (and the above item) right? If so, does the weird "rotation by 360 degrees causes a sign flip" thing show up in orbital mechanics anywhere?

5) The space of orbits (taking into account eccentricity, SMA, etc.) passing through a single point and ignoring orbital phase is the space of velocities in 3D Euclidean space (which is 3D Euclidean space itself), because for each velocity an object can have at a given point, there is a unique orbit passing through that point.

6) If we extend (5) by adding orbital phase as a coordinate, we get either:

6a) 4 dimensional space with the fourth dimension being cyclic if we restrict eccentricity to being less than 1 (because for each orbit that makes up (5) (minus the ones with eccentricity >= 1), we're considering all the points along that orbit, rather than just the single point we used to define (5).

or:

6b) Something a bit more complex if we consider all eccentricities, since the orbital phase coordinate will go from cyclic to acyclic when eccentricity is greater than 1.

7) If we consider all orbital elements (including orbital phase) without restricting ourselves to orbits passing through a single point, we get 6D euclidean space, because each combination of position and velocity corresponds to a unique combination of orbit and phase.

Are all of these identifications correct?

Are there any other interesting spaces relevant to orbital mechanics?

Does the space we get by taking (7) and removing orbital phase as a coordinate (i.e. the space of all possible orbits with the position of the orbiting object along the orbit ignored) show up anywhere else in math or physics?

 

[Thorsten]

4) If we extend (2) by adding orbital phase as a coordinate, we get SU(2). Have I gotten this (and the above item) right? If so, does the weird "rotation by 360 degrees causes a sign flip" thing show up in orbital mechanics anywhere?

Just on a quick glance, orbital mechanics is a classical problem - why would complex numbers and unitary groups appear here?

Your orbital situation is *the same* when you go around by 360 deg. There's no multiplication of amplitude and complex conjugate amplitude as in quantum theory which would allow a complex phase rotation to lead to a sign flip.

All distortions you can do to a closed orbit are either deformations in which a parameter changes linearly in a certain fashion or 3-dim rotations. The rotations around the origin leave the kinematics invariant because the background gravitational field is rotationally invariant, so you get the O(3) symmetry group from there. The deformations do not, because the field isn't translationally invariant.

 

[BLANDCorporatio]

Cool stuff. I'll try to give a few comments on each.

(Assumptions: Newtonian gravity; no upper limit on velocity; object around which the orbits are defined is a point-mass object at the origin which is also a ghost that one can pass through)

1) Yes, that is correct. (For anyone else watching, normals to the orbital plane are identified with directions from the origin, therefore there's a neat correspondence between orbital planes and elements of RP2 aka real projective plane, the real projective space for three dimensions).

2) Also true. Imo, you should have started with this It's much easier to visualize (one imagines the normals as unit length vectors, and the direction of the normal also specifies which way the orbit rotates, and obviously there's a sphere of unit vectors around the origin). In fact, one way to define RP2 is to start from the sphere S2, and 'identify' points that are antipodal: you can 'teleport' on this not-sphere-anymore from the north to the south pole for instance, because they become the same point. That's RP2.

I'd like to invert the order of answering 3 and 4.

4) This is not true.

What you suggest seems to be S2*S1, the direct product of the 2-spehere with the circle (to each 'point'/'element' in S2, you associate a circle and require an extra coordinate for the circle). You claim this is the same as SU2, aka S3 (the 3-spehere; SU2 and S3 are isomorphic, homeomorphic, diffeomorphic, 'the same thing' basically).

S3 is simply connected, which means that if you draw any loop on an S3, you can also continuously shrink that loop down to a point without ever leaving S3. The more mathematical way to state this is that the fundamental group of S3 is trivial (only contains one element, so the group is {0}).

It's easier to visualize this on S2, the regular sphere. However you draw a loop, you can shrink it to a point. (Its fundamental group is also trivial).

Oddly enough, you cannot shrink loops on the circle S1 (no you're not allowed to shrink the circle itself ). The circle's fundamental group is Z (the group of integers).

A property of fundamental groups is, if space A has f. group H, and space B has f. group G, then A*B has f.group H*G.

So S2*S1 has fundamental group {0}*Z, which is not a trivial group because it contains more than one element. It contains elements like {0, 0}, {0,1}, {0,2} etc, all of which are different elements.

OTOH, if S2*S1 and S3 were the same space, then they should have shared all topological properties including the fundamental group. They don't. They are not the same space.

Another way to state this, more intuitively this time, is that you can draw loops on S2*S1 that you cannot shrink to a point. However, any loop you can draw on S3 you can also shrink to a point, so the two are different spaces.

3) This is not true either.

What you suggest seems to be RP2*S1, the direct product of the real projective plane and the circle (because to each 'point'/element in RP2, you associate a circle and require an extra coordinate for the circle). You claim this is the same as SO3.

The method proceeds as above by using the fundamental groups of RP2, SO3, and S1.

S1's f. group is Z. A more intuitive way to state this is that a path that wraps around the circle clockwise once is 'different' than a path that winds twice, and different than one that winds three times etc. By different, I mean there is no way that you can continuously deform the first path into the second, without either breaking the path or taking it out of the circle.

For both RP2 and SO3, the f. group is Z/2Z also known as the Boolean group of two elements {0, 1}. Intuitively, this means a path that wraps around twice is the same as not moving at all (because it can be shrunk to a point), or a path that wraps once results in that sign flip you mentioned (because it cannot be shrunk to a point).

Using the same argument as above, the f. group of RP2*S1 is {0,1}*Z, which is not the same as the f. group of SO3, which is just {0, 1} and contains only two elements. Meanwhile, the f. group of RP2*S1 contains elements like {1, 2}, {1, 51}, {1, 103} etc, all distinct.

Since RP2*S1 and SO3 do not have the same f. group, they are not the same space.

5) This is dubious.

So let's assume the body you orbit around is at the origin, and you want the space of possible orbits at some arbitrary point P. By this, I think you mean the possible shapes of ellipsis (or whatever conics) the orbits passing through P may have.

However, one category of orbits have the shape of a line segment, of some length, aligned to the P-to-origin direction (the orbiting body falls into the origin).

So lets pick one such linear segment orbit, and on it lets pick some point not at the segment ends. Then, during its orbit, the object will pass through the point twice with different velocities (because one velocity is oriented to the origin, the other away from it).

If falling through the origin is too wonky to consider, then just imagine that at point P you give enough velocity directly away from the origin so that the orbiting body never returns. Again, you have a situation where at more velocities at P you have the same orbit shape (a line starting at P, moving away from the origin towards infinity).

6) Because 5, as far as I understand it, failed, this extension also fails.

7) I am wary of this 'identification' (aka, I think it is false).

Orbital coordinates include one coordinate that 'wraps around', namely orbital phase. Meanwhile, none of the directions in 6D Euclidean space wraps around itself. The two spaces are therefore topologically different and cannot be 'the same'.

Yes, I am aware polar coordinates exist. The simple example is polar coordinates on the plane, where you use a radius and angle to specify a position on the plane. However, what if the radius is zero? Then your angle coordinate can take any value whatever (an infinite of values), nonetheless you still describe the origin of the plane. There's a singularity there and therefore R*S1 (polar coordinates) and R2 are not isomorphic.

In your case, consider the 'orbit' of standing still at the origin. It's an orbit, and it corresponds to position=origin, velocity=0. In this case however the orbital plane is undefined.

Bottom line, and this applies to answers like 3) and 4) as well, there is a difference between the suitability of some space as a coordinate chart on some part of another space and the two spaces being 'the same'.

Final question ("does the space in 7 show up anywhere else"?):

With the proviso that the identification in 7 doesn't work, I'll take the question to refer to the space of possible shapes of an orbit. And the answer is yes, this looks very similar to action-angle coordinates and (foliations by) invariant tori.

I'm le tired now so I might explain what those are about in a future post, assuming someone else doesn't do it before me.

 

[Linguofreak]

2) Also true. Imo, you should have started with this

My thought process, at least involving (1) and (2), did, as I recall, but I used the order I did because (1) is in some sense "simpler" than (2), given that it lacks the specification of orbital direction, even though the space behind it is less familiar.

In fact, one way to define RP2 is to start from the sphere S2, and 'identify' points that are antipodal: you can 'teleport' on this not-sphere-anymore from the north to the south pole for instance, because they become the same point. That's RP2.

That is indeed how I learned to visualize RP2.

I'd like to invert the order of answering 3 and 4.

4) This is not true.

What you suggest seems to be S2*S1, the direct product of the 2-spehere with the circle (to each 'point'/'element' in S2, you associate a circle and require an extra coordinate for the circle). You claim this is the same as SU2, aka S3 (the 3-spehere; SU2 and S3 are isomorphic, homeomorphic, diffeomorphic, 'the same thing' basically).

The issue here is I *thought* I'd figured out that either (3) or (4) had to be the 3D rotation group, which I know is SO(3), and had figured out that (4) was the double cover (if I understand the concept of a cover correctly) of (3), so I assumed (3) must be SO(3) and (4) must be SU(2) (knowing that SU(2) is the double cover of SO(3)).

Before I realized there was a double cover involved, I had been thinking (4) was SO(3), because I thought I'd ended up with equivalent structures in trying to visualize them.

To start with, let's talk about visualizing the 3D rotation group. When I first heard about SO(3), I was confused about how it was different from S2. This is because a naive visualization of the group of 3D rotations is to take the possible positions the nose of an aircraft can have when the aircraft rotates around its center of mass. These positions describe a sphere. The visualization fails, of course, because it fails to take into account roll. For any nose position, there's a full circle of possible vertical stabilizer positions. It seems to me that any rotation corresponds to a nose position and a vertical stabilizer position at that nose position. Is this model correct? Does it miss some feature of the rotation group? My thought process was then that (4) seemed to lend itself to the same visualization, and so must be SO(3). Did I get something wrong there? When I realized that (4) was a double cover of (3), I thought that I must have gotten something mixed up and that (3) must be SO(3) and (4) SU(2).

S3 is simply connected, which means that if you draw any loop on an S3, you can also continuously shrink that loop down to a point without ever leaving S3. The more mathematical way to state this is that the fundamental group of S3 is trivial (only contains one element, so the group is {0}).

I'm familiar with the notion of a group being simply connected. I'm less familiar with, but I think I more or less understand it when I encounter it in reading (it's just that I forget about it between sessions of reading up on topology, so I keep having to look it up every time I encounter it).

Oddly enough, you cannot shrink loops on the circle S1 (no you're not allowed to shrink the circle itself ). The circle's fundamental group is Z (the group of integers).

Oddly? Seems self-evident to me.

A property of fundamental groups is, if space A has f. group H, and space B has f. group G, then A*B has f.group H*G.

So S2*S1 has fundamental group {0}*Z, which is not a trivial group because it contains more than one element. It contains elements like {0, 0}, {0,1}, {0,2} etc, all of which are different elements.

I guess the big thing I'm still wondering at this point is whether (4) is indeed S2*S1. S2*S1 has a circle associated with every point on the sphere, and so does (4), but it seems to me that SO(3) also associates a circle to every point on the sphere (from the nose/horizontal stabilizer visualization above), but seemingly in a different way since it's different from S2*S1.

To clarify what I mean about "a different way", consider the case of S1, S1*S1 (the torus), and S2:

S1*S1 takes a circle and associates a circle with every point on it. S2 also takes a circle and associates a circle with every point on it, but in a different way, such that a given point and its antipode on the original circle share the same associated circle, and are antipodes on that circle as well (Hmmm... the "antipodes" thing reminds me of the visualization of RP2).

BTW: Can we generalize the process that creates a sphere from a circle to other topological spaces? For example, the torus is S1*S1, so we can generalize that to other topological spaces as "taking the Cartesian product with a circle" (X*S1) or, more generally, "taking the Cartesian product of two spaces" (X*Y). Can we generalize making a sphere out of a circle to taking the Cartesian product of the circle with something (S1*X), or applying some special operation to the circle (S1* or S1*X again, but with * meaning something other than the Cartesian product)?

5) This is dubious.

<SNIP>

6) Because 5, as far as I understand it, failed, this extension also fails.

Fair enough. I kind of suspected I might be playing a bit fast and loose here. Is there any interesting known topological space (5) and (6) do correspond to?

Final question ("does the space in 7 show up anywhere else"?):

With the proviso that the identification in 7 doesn't work,

I don't think that should matter. The question was "does the space in 7, minus orbital phase show up anywhere else", and the subtraction of orbital phase from the picture is going to make that space look a lot different than 7 itself, making my misidentification of 7 moot (since I wasn't sure what 7 minus phase looked like anyways).

Anyways, I'm not too distraught to have misidentified the ones that I thought were fairly plain and Euclidanish, because flat Euclidean space (of any dimension) is boring. Does 7 itself work out to anything cool?

 

[BLANDCorporatio]

Interesting questions again. I won't quote from the post but I'll try to address them.

1) SO3 and its representations.

There was a question in a mechanics text-book: 'what is SO3? is it S1*S1*S1 (similar to but not quite Euler angles)? is it S2*S1 (axis-angle)? S3 (unit quaternions)? or something else?'

And indeed, the answer is 'something else'. The way to properly prove that is to show that some topological property (for example the fundamental group) is different for SO3 to any of the other proposed candidates.

The less rigurous, less definitive, but maybe more immediately useful way to hint that there's a difference there is to ask 'does this particular identification work?'

For example, 'does the axis-angle representation of rotation work to identify SO3 and S2*S1?' Note, if I show the identification fails, it could still be the case that S2*S1 and SO3 are diffeomorphic, you just need a different correspondence. (As said previously, S2*S1 and SO3 are not the same, and are not diffeomorphic, but whatever)

So what does it mean to identify a space with another, for topology purposes? It means you need:

- a bijective (aka invertible) function that takes an argument from one space and produces as a result a point in the other.
- that function must be continuous; very loosely speaking, if x and y are 'close', then so are function(x) and function; a bit less loosely speaking, it maps neighborhoods of x to neighborhoods of function(x) for any x. If this happens the function is a homeomorphism and preserves topology.
- (if you want to do calculus on the spaces) the function should also be smooth: its representation in any coordinate chart should be indefinitely differentiable. If this happens you have a diffeomorphism.

Our candidate function for today: the function that associates {a unit vector and an angle} to a rotation.

However, this function fails to be bijective: the null rotation means angle is 0, but the axis is undefined. Since any combination of axis with a zero angle maps to the null rotation, the function cannot be invertible, and fails to be a homeomorphism.

Another way to look at this is that you have a singularity in the axis-angle representation of rotation, and the singularity is at the origin of the rotation group. As it happens, it's a somewhat 'friendly' singularity: you have an infinity of ways to represent the null rotation.

But it's a singularity nonetheless.

As to what properties of SO3 are not present in S2*S1, I showed one in the previous post: the fundamental group. You can shrink any double loops on SO3. You cannot, for some loops, on S2*S1.

2) Direct products and how general they are.

Very general. You don't need one of the operands to be a circle. Any combination of spaces can be arguments for a direct product. RP2*SO3 makes sense, mathematically. I don't know where it might show up, but it is well defined.

And it is well defined because the direct product is, conceptually, very simple. Suppose you have a space A and a space B. Then A*B means that every point in A 'becomes' a separate instance of B. If you had some coordinates for A, and some coordinates for B, to specify a point in A*B you need first to select one point in the A component, and then select a point inside the B-subspace associated to that point.

Also, and this is very important, all the points you specify this way are distinct.

Example: suppose you have R*R, the plane. Then there are no (x, y), (a, b) pairs that have x different to a and y different to b that are the same point, nonetheless.

Let's now take your 'other way' into consideration.

As I understand it, you start with a circle. Pick two opposite points, and pass a circle through them, perpendicular to the first. Rolling this circle around an axis perpendicular to the plane of the first circle produces the sphere.

However, as you yourself notice, this construction associates a circle to a pair of opposite points. Also, all circles pass through the same two points; call them North and South Poles, if you like. Unlike the direct product, a lot of the points you create are 'the same'.

Can this construction be generalized? I think it can, it seems a straightforward [ame="http://en.wikipedia.org/wiki/Surface_of_revolution"]surface of revolution[/ame] kind of thing. Given enough dimensions to embed the original surface/space and the result, you should be able to pull it off for whatever shape you wish. Note that surfaces of revolution are done by rotating something, so your operation has only one argument that's a surface or a space, not two as the direct product. Surfaces of revolution also, in general, are not direct products.

3) What am I doing when putting more coordinates.

A direct product.

If you have a space, and then decide that you need to specify, for each point, some kind of value that itself is a point in another space, you are doing a direct product.

You can later decide that actually some combination of values are 'the same', which means you 'glue' points in the direct product space, but that changes its topology.

A typical example: [0..2pi]. You decree that 0 and 2pi are the same, and because you glue these two points, you get the space S1. Obviously however, a line segment and a circle are topologically different. In particular, one has a boundary, the other does not.

In your case, let's say you specify some unit vector (a point on S2) and an angle (a point on S1). Without knowing anything else, every combination of axis and angle is 'valid' and distinct. It's only when you smuggle in more info (zero-angle is the same rotation, regardless of axis; (angle x and axis v) is the same as (angle -x and axis -v)) that you get SO3. But that info isn't smuggled in just by putting more coordinates to the problem. It's smuggled in by gluing points.

4) What do spaces in OP 5, 6, 7 correspond to?

Like I said in the previous post, the most interesting correspondence I see is axis-angle coordinates.

One way to represent the state of a physical system is phase space: you need some position (aka configuration) coordinates to tell you where it is, and some momentum coordinates to tell you, essentially, what it is doing.

Without any other information, phase space is 'boring'. You have n position coordinates (which are often angles), n momentum coordinates (which for now I'll assume unbounded) so phase space looks a lot like S1*S1*....*R*R*...

A physical system obeys some dynamical equation, which means that it defines a vector field on phase space: to each point in phase space, it associates a direction (and speed, but whatever). In other words, if the system happened to be at a given configuration with a given momentum, the vector field will tell it where to go next. You could place the system at some point, then trace its evolution in time. You could in principle do this for every point in phase space. In general, the shapes of the trajectories will be fairly complicated.

However, at least for some systems, it is possible to 'warp' the coordinates on phase space in such a way that you get a new set of momentum and configuration coordinates, which have the following property: the momentum coordinate is conserved along a trajectory of the system. That's the 'action' part of action-angle coordinates.

(Fundamentally, momentum is conserved, so if your system description is complete enough, there should be a way to find constants of motion to give you the conserved momenta. You can however formulate systems where it's not obvious what momentum is conserved: how about a pendulum in uniform gravity, or a perfectly elastic ball bouncing on the ground with no energy loss? Without considering the influence of the pendulum or ball on the Earth, you have this pesky external force that keeps adding and removing momentum to your system)

This constrains the shape of the system's trajectory to what is often topologically a torus, a direct product of several configuration coordinates that are often angles. You might then be able to find how fast the system goes around, aka its frequencies, for those angle coordinates.