Oh, this is a fascinating conversation!
Here's my 2 cents (may be worth less, I don't know!)
For me, collection is (as others have suggested here), an energetic arranging of the body that translates into a certain quality of movement -- one that is extremely balanced, pulling the center of gravity inward (often upward, but I don't think only) so the body can move in any direction with power, suspension, and lightness. I think it's like a rubber band, as tension (for want of a better word -- relaxed tension?), extending out and coiling in, with a strong sense of that centered balance coming from the core of the body.
I think the position of the body comes from this energy, and that we train for position to help find the energy, if that makes sense! I think it's why the "frame" is so seductive -- by putting the horse into that position, we think collection with follow, but it doesn't automatically. But the position (or positions) can help the horse find a way to find collection and balance. So it's a dance between the two all the time, I think.
Speaking of dance, I'm going to throw a spanner in the works here and turn left and use the human body to illustrate my point. In a former life, I was a professional dancer, and it really struck me when Josepha mentioned somewhere (sorry, I'm not remembering where -- posts are blurring in my head because I've been reading so many!) that in her perception, it happens in many instances, even with people, and used dancers as an example. This made so much sense to me!!!
So I thought it might be interesting to look at human dancers to think about equine dancers.
First, I wanted to share a concept from ballet: ballon
Ballon refers to the ability of a ballet dancer to appear to hold a position in the air. It may be named after the French dancer Claude Balon, however it may also come from the French word ballon, meaning "balloon". It also refers to the quality of jumps/leaps or allegro steps that appear to have a soft but sustained rebound from/to the floor or stage, for example, a series of balotte saute. A dancer with good natural ballon usually has great natural spring and long achilles tendons.
To perform the more demanding routines, a ballet dancer appears to defy the laws of physics. Basic physics and understanding of human perception provide insight into this. A high level of physical fitness is required.
For example, during the grand jetÃ©, the dancer may appear to hover. Physically, his/her center of mass describes a parabola, as does any projectile. Observors have limited ability to reckon center of mass when a projectile changes its configuration in flight. To do this the dancer extends their arms and legs, which camouflages the fall and leads the audience to perceive the dancer is floating.
â€¦The landing must be performed carefully, the dancer bends at the knees (plies) and rolls the foot from toe to heel.
For me, this sense of ballon is the equivalent of the suspension a horse finds in true collection and extension, rather than a specific head set or angle of hips or alignment of the body. The deep bend of the knees in piaffe or passage is to me the equivalent of the dancer's plie, building the spring.
One of the greatest dancers of our era, and exhibitors of ballon is Mikhail Baryshnikov. Although only about 5'7" or 5'8", he could, at the height of his career, jump six feet in the air. Think about that! He could jump higher than he was tall!
This is a clip from the movie White Nights, of him dancing to a piece called "Fastidious Horses" (:-)). Here he's dancing with extreme collection, and a great deal of tension/drama -- he's sharing his despair over what it felt like to be trapped in Russia and what he perceived as the tightness of classical ballet and being an artist there, and his desperation to find the freedom of expression his character sought by defecting. (So, this is not a relaxed collection, but it's an amazing visual of what extreme collection can do, to my eyes).
Here's a clip from a Twyla Tharp piece, that is much more relaxed and lyrical. You can see him move from a very loose sense of collection (very typical of Tharp's modern choreography), and then watch him pull it together as he begins to move farther into the piece. And, I think there are great examples here of what Donald has been exploring about the extension into the jump of horses -- I think he does the same thing:
(I also think this is interesting because the quality of his movement is so different from the ballerina he's dancing with -- he looks to my eye like a Spanish horse or Lipizzan, compact and springy, and she looks like a Thoroughbred or Arab, much lighter.)
And here's another clip from White Nights, where he's dancing with Gregory Hines. I think this is intriguing because they're dancing in unison and are both fine dancers who come from very different training traditions (Hines was a tap dancer) and do the same movement so differently. They are both collected, but Baryshnikov is more so -- Hines is looser, his center smaller and lower to the ground -- I'm thinking that there's a parallel here between dressage horses and western horses.
Also, watch the very beginning as they begin to walk into the dance -- they do it twice, and you can see them pull themselves into collection twice.
http://www.youtube.com/watch?v=haBZCrBH ... re=related
And here's a short clip of him teaching a master class in Cuba to a group of young male ballet dancers. I think this is really interesting for a couple of reasons. First, most of the time he's standing, watching, not in any collection. Then he'll react to their lack of collection by flinging himself around, and then demonstrates what he's looking for and pulls himself into collection. You catch a couple of glimpses of the young dancers, who are working to find dance's version of true collection, but you can see the difference in the quality of movement. When Baryshnikov does the movement, it seems to emanate, effortlessly, from his core. When they do it, they're far more concerned with pulling their bodies up into an artificial posture (a dancer's version of a "frame"). He's trying to push them past this.
http://www.youtube.com/watch?v=BgzRQg6k ... re=related
I'm sure that Donald could do this same process with martial artists!
Lastly, here's an article on imagery and the kinetics of jumping for dancers from a book by Eric Franklin. I think there are links here for both how we can imagine collection in our horses and the biomechanics involved. There are interesting ideas about muscle types, alignment, and the release of tension:
http://www.humankinetics.com/products/s ... pt_id=3539
So, I guess I've not done anything to specify the minute "what's" of a horse's collection, but I hope this is interesting/helpful! The way my brain works, when I'm having trouble landing on something directly, it can help to look elsewhere to explore the idea as a metaphor from the outside in. And I don't think it's accidental that we refer to horses as "dancing" when they are moving in collection!
All the best,