Dressage is the only Olympic event that can claim Xenophon as its first coach. Photograph by Tereza Červeňová for The New Yorker

The piaffe is probably the most demanding and exquisite movement in the Olympic sport of dressage. A horse in piaffe defies what horses otherwise do. Instead of going anywhere, it jogs on the spot, three-quarters of a ton of moving muscle, feet rising and falling in the same four hoofprints like an animation in a flip book. Next week, in Rio de Janeiro, seven judges around an arena, known as a manège, will evaluate the piaffes of the four-day dressage competition. In addition to making sure that the horses don’t go forward or backward, or side to side, the judges will keep track of the number of steps (twelve to fifteen), their height (as high as the cannon bone on the foreleg; as high as the fetlock on the rear), and insure that they are not, in the somewhat baroque language of the sport, “unlevel.” Then they will score each piaffe out of ten.

No one knows what piaffing is for. The movements of dressage are said to have their origins in the training of horses for war, and one theory suggests that the piaffe might have been useful for trampling enemies. But the piaffe became an abstraction long ago, like the pike in diving, or the asymmetrical bars. By 1733, when François Robichon de la Guérinière, the equerry to Louis XIV of France, wrote a seminal guide to horsemanship, the piaffe had already become a thing of mere ornament. The correctly piaffing horse, de la Guérinière wrote, “stands in awe of the rider’s hand and legs.”

Charlotte Dujardin, a thirty-one-year-old British rider who is the European, World, and Olympic dressage champion, rode her first piaffe in the summer of 1999. She was in a sand arena at Wrotham Park, a Palladian manor in the suburbs of North London. Dujardin, who was fourteen, was spending a week helping out Debi Thomas, a friend of her mother’s, who worked in the stables on the property. Thomas had a twelve-year-old dressage horse that was trained to Grand Prix level—dressage has eight “heights,” of which Grand Prix is the highest—but it was struggling for rhythm in its piaffe. She had been schooling the horse, a mare named Truday, from the ground but needed a rider on top.

Thomas has trained horses for forty years, and she has never, before or since, put a child on the back of one trained to Grand Prix. Dressage horses are frequently compared to gymnasts. From the age of four, they undergo five or six years of strengthening and suppling exercises before they’re able to carry out the advanced movements: the piaffe, the passage (a slow, prancing trot, pronounced as in French), and the pirouette (a hand-brake turn, ideally executed in six to eight strides). There are fewer than a hundred Grand Prix horses in Britain, and a good one costs several hundred thousand dollars.

But Thomas had been watching Dujardin ride since she was a toddler. Dujardin’s mother, Jane, used to keep a pair of jumping horses at home. At the age of two, Dujardin would scramble onto their backs and gee them round the stables, clicking and hollering. When Dujardin got on Truday, and followed Thomas’s instructions—shortening Truday’s strides, shifting its weight to the hindquarters—the horse began to jig. “It was no big deal to her,” Thomas said. But Dujardin looked down and caught the expression on the trainer’s face. “She was, like, mesmerized,” Dujardin recalled recently.

Within days, Thomas had Dujardin performing flying changes—in which the horse skips from one foot to the other, in mid-canter—and the passage. “She just explained what I needed to do, and that was it,” Dujardin said. Her mother looked on from the rail. Jane had grown up on a farm in Hertfordshire. She had been an ardent show jumper, but her parents never came to watch. When Jane had children of her own—two daughters, Emma-Jayne and Charlotte, and a younger son, Charles—she poured herself into the world of show ponies and junior competitions. The girls began competing at the age of three. “They did want to do it, because it was my passion to make them want to do it,” Jane said. The family kept only first- and second-place rosettes.

“Literally all our money went on it,” Dujardin’s father, Ian, told me. “All of it.” Ian ran a packaging company, and in 1992, when Dujardin was seven, he won a large contract to wrap up mirrors. He spent fifty thousand dollars on a show pony for his daughters. But by the summer of 1999 another packaging deal had gone badly wrong. “It pulled everything down,” Jane said. “Our house, our home, our everything.” The Dujardins had to sell the show pony, and the horse box.

As Jane watched her daughter ride, she felt both joy and dread. It was obvious that Dujardin should pursue dressage. There was just no way to afford it. Even within the expensive world of equestrian sport, dressage stands apart for the aristocracy of its ideals and the wealth of its participants. Ann Romney sent a horse to the 2012 Games. In 2008, Denmark was represented by Princess Nathalie, of Sayn-Wittgenstein-Berleburg. Élite foals cost as much as sixty thousand dollars; medal-winning horses go for millions; the expenses of taking part are fantastic; and the prize money is pitiful. The careers of top riders can last decades, so the best horses and the richest benefactors have a way of gravitating to them, concentrating the glory of dressage like the blood of the Hapsburgs. “It’s a vicious circle,” Astrid Appels, the editor of Eurodressage.com, one of the sport’s leading Web sites, told me. “The weak in the wallet can often not afford competing at international level.”

The Dujardins knew all this. “It has always been a pompous sport, a money sport,” Ian Dujardin told me. “We were just about paying the rent,” Jane said. “How was I going to fulfill what I thought she needed to do?”

Dressage is one of those Olympic sports that you catch yourself watching when you walk back into the room and realize that you left the TV on. It’s legacy stuff, like archery, or the hammer, that sneaked into the Games at some point and hasn’t quite been thrown out—although dressage has come closer than most. At the 1952 and 1956 Olympics, blatant favoritism by judges to riders from their own nations almost led to the sport’s expulsion. (Prince Bernhard, of the Netherlands, intervened.) The solution, which involved filming each ride and arguing about it for half an hour, pretty much killed dressage as a spectator sport. “The interest of the public died down alarmingly,” Colonel Alois Podhajsky, the director of the Spanish Riding School, in Vienna, wrote of a visit to the Rome Olympics, in 1960.

The sport was rejuvenated in the nineties, when a new event, the freestyle, came on the scene. The freestyle made its Olympic début in Atlanta, in 1996, and since then has helped nudge the sport toward the same emotional, aesthetic realm as figure skating: no one knows what the hell is going on, but at least it looks nice. In the freestyle, riders devise their own routines, which are set to musical medleys, usually with any words removed, because they can distract the horses. The event is the climax of the Olympic competition and decides the individual medals. During the previous three days, horses and riders compete for team medals in the sport’s traditional tests—the Grand Prix and the Grand Prix Special—which involve a strict series of movements. Then everyone watches novel combinations of piaffe, half-pass (in which the horses go forward and sideways at the same time), and extended trot, while the theme from “Pirates of the Caribbean” blasts across the manège.