In Flight, to prevent the plane from diving, the pilot puts it upside down before flipping it back over just before landing. This is supposed to be loosely based on flight 261 from Alaska Airlines which unfortunately crashed without survivor after attempting a similar desperate manoeuvre. Let's just focus on the inverted flight part. Can a commercial airliner (50+ passengers) fly inverted for more than a few seconds before aerodynamics and/or propulsion problems arise? What are these specific problems?
Absolutely. As a matter of fact, one of the first commercially successful aircraft (the Boeing 707) did a barrel roll during a demonstration flight. In case you don't believe my claim, here is a video to back up my claim:
(EDITORIAL NOTE: I take exception to the part where the NatGeo writers state "something the aircraft had never been designed to do" because a perfect barrel roll is a constant 1G maneuver (generally with anything between .5G to 3G, well within most aircraft design parameters), and doesn't stress the aircraft in any way, although this particular execution was probably not per any aerobatics manual due to the actual fight going on.)
Basic aerodynamics doesn't really care what the orientation of the aircraft is as long as you are able to balance the forces. The basic forces you have to deal with are weight, lift, thrust and drag. In an inverted position, you may think that the combination of lift and weight would be detrimental to flight, however you can tilt the angle of attack of the wing so that even in an inverted position you can get the lift to be opposite of the "top" surface of the wing. Otherwise an aircraft would have a hard time descending.
The Smithsonian Air & Space museum has a nice web page that explains how inverted flight works. Aircraft are basically the same on a meta level. The thing to keep in mind, that many subsystems of the aircraft may not operate well for extended inverted flight. However, (and keep in mind I have not seen the movie in question) most systems will maintain operation for at least some time due to residual fluids and such in the oil and fuel systems. Depending on the aircraft make and model, the fuel and oil lines should provide at least 30 seconds to a minute of uninterrupted operation.
For aircraft that do fly inverted, you can check out any airshow and see all manner of aircraft fly inverted for extended periods of time. Jets and propeller aircraft have well understood physics. You can learn more about aerodynamics at this page if that helps. If you look at this image:
All you have to do is get the angle of attack to be negative for the right-side up wing (thus if the wing is upside down, it's creating a net up force against gravity). So again, it's quite possible and doesn't violate any flight dynamics.
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Yes, you can fly an airplane upside down, as long as the airplane frame supports the unexpected loads and the fuel, engine oil and battery acid stay where they are supposed to.
Contrary to popular belief, a plane flies not because the air above the wings moves faster than the air below, but because of circulation. This means that there is no reason why a plane couldn't fly upside down, as airshow pilots routinely do.
The links point to a very good book by John Decker on how to fly airplanes. It explains quite thoroughly why airplanes behave as they do.