Ughhh, I can't stop flapping like a bird!!

Ok well my balance at the canter has much improved and as you will notice in the video, I can now do a smooth, relaxed transition from the walk to the canter. However... MY ARMS!!!! Uuuuugh. I look like a flapping dodo bird! ANy tips to get them still?
(I think (I think) that one of the reasons they flap is because I sort of move my hands to follow the horse's head), but my reins are loose enough that I don't have to do this, so I think next lesson I am going to try to keep my arms/hands as still as possible). That would probably help but I would like some opinions. Also my position isn't always very pretty. I feel very secure and my butt stays rather well on the saddle (if we exclude the beginning of canter and the rare irregularity), but my upper position is just... well watching the video will say everything.
I think I might have to pull my chest/belly button more upward and curve my back a little. But then again, I'm not sure so please tell me what you think!

If you notice something else don't be shy to say it. I don't mind very honest critiques.

Hmm...does she seem lazy to you? It seems like you're trying to 'pump' with your upper body to push her forward and stay in a canter. Instead of pumping with your body, you should just put your leg on her.

Hmmm, now that I think of it... that too. She likes to suddenly stop cantering and burst into a very fast trot if I don't really encourage her with my seat. But I will try just giving her more leg like you say instead.

Try just sitting deep and closing your leg on her to urge her forward. Carry a crop, often that's enough and you don't even need to use it.
I only watched a short segment of the video, but I think that's what's happening with your upper body. It's a really common mistake, but fixable! :)

I was taught that your arms should move a bit with the horse as long as you are not pumping. But that would mean that your body stays still, and only from about your elbows to your hands are moving...and its more of an up and down motion as your horse goes up and down with his stride.

Have to agree with spastic that it looks like you are pumping your body to keep her going. Second the carrying a crop. Ask with the leg, then tell with the crop if you got no response from asking with the leg. Don't "nag" at her, but make her listen. Really good excercise to start practising at the walk first, then the trot, then the canter.

I think part of it is that the mare is strung out a bit and heavy on her forehand. If you collected her up a bit, sat up tall, and take a bit more in your hand and leg, you could keep her going without having to pump the same way. Its like she's pulling herself along with her front end, instead of pushing with her backend.

Yeah, she's got you working harder than her. Upper body still, more leg, crop to re-enforce leg. Always go back and ask again with quieter aids if you have resort to crop to get her in front of your leg.

I noticed that, and I already thought that she would be so much nicer at a more collected canter, but how do you collect???

Ya I had a crop in the beginning but she was doing well and I wanted to try to get her to canter by just asking with my leg. She did that beautifully but I will keep it in the future so I can use it to keep her going more. I already squeeze her with my legs to keep her going at the canter and I can feel she accelerates a little...

Ok, so I read up on collecting and basically you ask more impulsion by squeezing/encouraging with leg cues, and then you gently do half halts/tug a little to get the head to go up. When the horse gives a little, you release, and then ask for a little more a few strides later until his head has gone up enough. This all right?

Yup she looks like she needs constant leg to keep her going, I agree with Spastic_Dove, leg first crop after, and don't nag at her TELL her.

It appears you are following the canter motion with your upper body instead of your hips. You seem tight in your low back.
The elbows popping out to the sides can be fixed by hanging your upper arms straight down & placing your hands on your upper thighs with the fingers pointing towards your knees & your thumbs in the area where the tops of your thighs meet your body. This will also allow you to move your hips more readily as it will be harder to move your shoulders so much.
Of course you'll look kind of dorky (but that's OK) & it's best to do this on a longe line unless your horse will stay in gait & go with minimal guidance. If those won't work for you, you could try it one arm at a time although that may push your non-thigh shoulder forward.
You have a nice seat & legs.