How does this chair move? I have a chair like this(shown in figure). I was kneeling on the part where we sit. I just gave a sudden push to the part where we rest the back, without moving my legs. I found that the chair is moving little in the direction I applied the force. I couldn't find any reason why this chair would move forward because I am inside this system and no external force is coming from outside. When I apply the force on it , equal amount of force I am applying there on the seat in the opposite direction, so there should not be any movement.
I tried this experiment again right now. Now holding on the handles of the chair. Did the same thing. Again found the chair moving.
Where I am wrong and how does the chair move?

 A: In free space, this would be impossible, but there's friction, so if you impulse the chair forward, by moving yourself a little back (by conservation of momentum the chair must move forward), and then you get back to you initial position, the chair will move back again, but there's friction and energy is being lost during the process so the distances will be smaller and the chair, at the end, will be a little bit moved forward.
A: Let’s assume that you are on a frictionless surface along with your chair. Now let’s consider the chair as a system. When you applied the force, you are a different system: hence your force is  an external one. This external force moves the chair, according to Newton’s laws.
Now for your question about being inside a system. If you think of yourself and the chair as a single system, you would have moved due to the reaction force from the chair, which I guess you didn’t, because frictionless surfaces are hypothetical. If you were on a frictionless surface, both you and the chair would have moved due to mutual action and reaction, but your center of mass would not have moved.
I guess you are thinking of internal force force as something from inside a body. Well, any force which comes from something not lying on your system is an external force. In the case of your chair, your body (as it is a different system) is another system exerting the external force. However, the force that the molecules of the chair exert on each other are internal forces: they can move individually due to their forces, but as a whole their movement is in such a way that the center of mass of the chair does not move. The motion of the center of mass of the chair is what is required for the translational motion of the chair to be perceived.
A: A certain minimum force is needed to move the wheels. If you move your body slowly, you can move forwards (or back) on the chair without the chair moving. But if you make a sudden move, the force you exert on the chair (and that the chair exerts on you) is greater than this minimum.
This means you can move by using a different acceleration in the forward direction compared to the reverse direction. You can enhance this effect more by "jumping" with the chair - if you pull up on it (or if you push off the chair so there is a time your downwards force on the chair is reduced) you again lower the force needed to move it.
It is possible (but no necessary) that the difference between dynamic and static friction plays a role - things that are moving typically experience slightly lower friction than things that are sitting still. Thus is you move gently while the chair is still, the chair will remain stationary over a wider range of accelerations than you might expect.
The following diagram might clarify this:

The blue line represents acceleration: when the acceleration exceeds the "minimum" required, the center of mass will barely change (chair moves back, person moves forward). When you move slowly (below the critical acceleration) the center of mass can move (there is a net force on the system from the floor - a small force, but acting for a long time).
