Position of the helicopter

You need to define what "stationary" means. Nothing is every really stationary...it's all moving around he sun, the sun moves around the galaxy etc..

If you tell the pilot to remain stationary above a point on the earth he will still be there the next day. If you tell him to remain stationary with respect to the sun he won't still be there (or even on the planet if his helicopter worked in space!).

 

Perhaps also see..

http://en.wikipedia.org/wiki/Foucault_pendulum

 

If we ignore problems with the way the question is set then yes the helicopter might remain fixed while the earth rotates below it. Much as the moon remains facing the earth as the earth rotates below it.

PS It really does matter how the question is set. In practice a helicopter can't remain at a constant height without the rotor remaining roughly parallel to the earth surface.

 

Remember that the helicopter and atmosphere of the planet rotate with the planet. A helicopter won't be "left behind" when it takes off. It will continue to rotate with the earth.

The more interesting question is this.. Does the rotor act like a gyroscope and try to keep the plane of rotation constant leading to the helicopter "falling over".

 

You can try this.
Inside a moving train, throw a ball vertically and without moving the postion of hand, will it drop at the same position as it left your hand.

If return to same point, then you must ask what make it follows you?
Once it tossed up, it left your hand and the train and not connected with anything.
It is all about vectors.