Excersise 1: the basic volte is a small circle usually 6, 8, or 10 meters. The primary difference between riding a 20-meter circle and a volte is that bending of the horse's body is the first priority on the larger circle, while turning is the first priority on the smaller one. Bending creates an inside and an outside , which encourages the horse to "fill out" the outside aids. Then, the outside aids (hand, knee, thigh and calf) can turn the horse. This should not be confused with neck reining I don't see how it would be considering the way you hold the reins is a big difference! Well, if they're talking western neck-reining. But that is regardless.... In the volte, the bending aids must be so well-established that the horse retains the bend when the rider uses the outside aids to turn him.
The paragraph goes on about the technicalities of the volte, but I'll try and summarize with how I interpreted it (which might be totally wrong!)
The circle is so small that the horse cannot efficiently turn unless he actually bends as a whole. A larger twenty-meter circle would have him not bend so much, since he is not making that tight of a turn. The larger circle just teaches him to listen to your turning aids, which will become more and more defined and exact as you move up in the levels. The tight turn of a volte exercises the muscles that will later help him to collect himself, because in the tighter turn, the horse must step under himself more to control their hind thrust and they must collect their front end - the two-wheel steering, so-to-speak - so they make that tighter circle. The volte, in general, is used as a collection exercise.
It is hard for a horse until his muscles get used to being used that way. It is different then the forward-moving trot they've usually been asked for at training level. Just try turning in a tight circle yourself for a little while, and you might feel a little crick here or there. But it is not bad for a horse, it is just toning up different muscles. And you'll need it later on!
Why do people usually refer to a general horse as a "he"? Just curious...
Anyway, I actually had to use my crop on Greta! It's brand-spanking new and really nice, but Greta has never needed for me to use it. The voltes needed some extra encouragement, and squeezing as hard as my skinny little legs could wasn't cutting it. I never had to whack her, just a light little tap, and then she listened for the rest of the way once she saw the crop in my hand. You're not supposed to whack 'em with it anyway unless they've repeatedly not listened to your aids. I always make sure I'm conveying my aids right before I resort to a firm tap. I wouldn't want to be punished for simply not understanding something!
Greta has gotten the hang of a 10-meter volte at the walk and trot. We could use a little more work on the trot, and then we'll try a canter. After we can perform a 10-meter volte very nicely, then we'll go onto 8, then 6. We have large cones in the arena that do help with judging the distance, though I do need to do some actual measuring!
Enjoy the little video that contains some footage of us "volte-ing" and the nice trot Greta had going on afterwards. The videos of us trotting before the volte are actually of her trotting after the volte, at about 0:36. The videos of us actually "volte-ing" start at 0:50. Once again, enjoy!