Okay, you need to use your arms. Right now they're kind of floating (I'm ignoring the trainer) there needs to be an involvement of your arms. To GIVE to the horse, you open and close your elbows. When the horse reaches(or bobs) down your elbows extend, when the horse comes back up, they close.
You can even practice this sitting in your chair. Thumbs on top, follow the horse's movement. Now as you get better, you will do it from muscle memory but for now you have to remind yourself to be nice and fluid and supple (Jane SaVoie's favourite word :P) But when you ride at the trot, your arms stretch out down, when you sit, they close again. Otherwise you yank your horse in the mouth and not only is it kind of painful but it teaches your horse to evade the bit and not really get comfortable. When you walk, same thing. Open and close your elbows as they walk with that head bob. Same thing at the canter.
If you don't follow with your arms, then your horse will bob and hit the bit and he won't keep impulsion, he will start evading the bit via pulling the reins or going behind the vertical or gaping his mouth.
So once you get those arms sorted, then you'll be able to focus on your core. Your seat bones and your core are what you use most when riding. You can ride without arms and legs, they are important but without having a good core and using your seat bones they're kind of useless.
You use your seat and core to communicate with your horse. Your seat drives the horse forward, to steers the horse (as you become more sensitive about weight changing) and it cues the horse along with your legs.
So when you aren't riding, work on your balance and any kind of ab workout and you will be able to accomplish much more.
I want to see you stretching your legs down, keeping your arms close to your sides but willing to stretch and bring them back as necessary (regardless of rein length,) and your shoulders need to roll back and push down, opening up your chest. Your back needs to be straight, not arched out. When you hollow your back, your horse will hollow theirs since you aren't nice and supple and shoclk absorbing.. you actually start tapping or slamming down on your horse's back (especially at the sitting trot and canter.)
Good job looking up, that's one of the things I really mess up haha!
Now with all riding, you're working on keeping the horse bending and not bracing. You want to use your inside leg (I usually tap before I squeeze) to ask your horse to move into your outside hand. On greener horses I like to position my outside hand a little farther from the horse's neck. Your outside leg is a supporting leg. It has different roles depending on what you're doing, as you should know from riding in general :) hopefully!
Around 0:40 were you trying to sit the trot? I have sound muted so I can't hear the trainer or disturb anyone here.
See how your arms are yanking him around. If you work on your arms, they'll become quieter. Work on keeping your upper body quiet, and absorbing all of the movement in your lower half. You're probably bracing a little bit, which is why your upper half is loud.
You'll work on that over time :) But exercising off of horses will help tons! Yoga, Pilates, any exercise ball work, hiking, running, etc.
You need to sit back and drive the horse with your seat and your legs, right now you're leaning on your hands and bracing against the stirrups. Do you see that?
0:57-1:00 is great, see how your body is following the horse, your arms are following, your body is stretching up and your back is flat. See how loose your hips are? I think you were trying to trot and your horse wanted to walk, but that looseness is what you want all of the time.
Notice how after that lovely 1:00, your horse begins to lose impulsion and he isn't going his usual speed. That's due to the reasons listed above (bracing, bopping him in the mouth, loud on top, etc.) so you'll notice when you begin to get more comfortable with that sitting trot, he'll keep his impulsion :)
You'll get there :) You have a lot of potential!
1. Work on your abs and balance off of the horse
2. Practice using your elbows to give and take from the horse as to follow its movement
3. Practice sitting on an exercise ball and keeping your back nice and flat
4. Maybe ask your instructor to put you on the lungeline so you can practice riding without hands
Hope I helped, I'm beginning too and learning lots (and currently not with my horse atm).
"Strength is the ability to use a muscle without tension"