I need some help with getting my driving gelding to speed up in the trot.
I had the same problem both with my green horse and my "made" show horse. I discovered they had much the same issues as my ridden horses, so I tried the same things on the drivers as I do with the riders, and it really worked. Here are a couple things to try:
Make sure your horse has the physical muscle to extend his trot. Hill work - as much as possible. And make him
walk up the hills. At the walk, the horse has to use each leg individually; working them at a walk, uphill, builds them up.
Make sure your horse is balanced at both speeds of the trot he now has. Is he working off his back end? Lots of transitions walk/halt/walk, then walk/trot/walk, then finally walk/trot/halt/trot/walk. Also work both trots and the walk on turns, circles and serpentines - both directions. These exercises teach the horse to work with his quarter under him, help him learn to balance on his quarters - both front to back and laterally - and build a muscle structure from which he
can extend.
Trot
down hills. Once he has physical structure and balance, he will have more faith to lengthen the trot and downhill work will build confidence. BUT!! When working downhill, don't just let the reins loose and tell him to yahoo along. Maintain a feel on his mouth. If he feels like he's just hanging on the bit - balancing on his forehand and your rein - give him little checks back. If you can, check back in the rhythm of a hind leg, and at the same time continue to encourage him forward. The idea being that rather that falling on his forehand, you are now bringing his quarters (that you have worked so hard to build up) back under him and into gear. If you check too hard, you discourage the pace, so consider that.
My "made" horse has shown for about 7 years as a pleasure horse. He had very definite ideas about what was "correct" which meant we had to tear down some walls. This was impossible in his EE cart, I believe because it was what he had been trained in. I put him to my HyperBike (which he considered absolutely NOT "correct) and he was more willing to try new things - like an extremely extended, but very cadenced and rhythmic trot. We have worked on his extension now in both the 'Bike and the EE, and he will work equally well in both. A couple weeks ago at a local show, we went in an open driving class. The class had 4 minis, a Quarter, another horse, and a Friesian. The judge asked for us to "Show them OFF!" and the arena was large. Rascal gave her not only an exceptional transition from road trot to rhythmic extension, he actually passed the Friesian - TWICE! People on the rail started yelling and cheering him on. I know the Friesian driver was a little miffed at placing 2nd to a mini, but even had to laugh. He told me, if you didn't see anything to give scale, Rascal looked like an 18h steam locomotive - and he hoped it would shame his Friesian into picking up his extension.
Take your time building up the speed, get a good foundation and built confidence. The speed will come, you will be awesome, your horse will be - and look - uber powerful!