newtonian mechanics - Internal forces in an isolated system

I did a weird activity and now I am trying to figure out the physics behind it:

While sitting on a chair I kept my feet above the ground and tried to move the chair forward. I was able to. Initially, I thought that perhaps the center of mass won't be changing as I was pushed backwards. But I moved the chair for a while and saw that it had moved a considerable distance and I was still sitting on the chair. Hence the center of mass did change for sure. According to Newton's second law, there must be an external force acting on the system(me and the chair). But I am not able to figure out what force is it. I have a feeling that it is somewhere related to the fact that the system is not completely isolated but I am not sure how.

Note:
I am not in contact with the ground or any other object close to me. I have lifted my feet and have my hands on my lap.

Answer

The movement you are performing is similar to hopping or jumping - see diagram below. It is much more difficult when you sit on the chair than when you stand, because you do not have as much flexibility, but it is essentially the same kind of movement. You could do even better by standing on the chair and tying it to your feet, then jumping on the chair as you would on the ground.

Using your arms (instead of your legs) as springs, you squat in the chair (b) then suddenly push yourself up and forward, dragging the chair with you when your arms are extended. It is slightly more effective if you hold yourself upright first (a) then swoop down before pushing all in the same movement. Swinging your legs also helps - when jumping you swing your arms and push with your legs; in the chair you swing your legs and push with your arms.

You could think of the chair as shoes - you push the ground through them, and because they are tied to your feet they leave the ground with you when you jump.

There are external forces acting here - you and the chair are not an isolated system. Contact forces with the ground - friction and normal reaction - as well as inertia/momentum, are essential - you cannot perform this kind of movement in space or when floating in water. You have to push down and back on the ground in order to get the ground to push upward and forward on you.