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As suggested by a user in this question, I'd like to ask this question:
One of the most basic aerobatic maneuvers is a vertical climb. This can be achieved by just pitching up to what appears to be 90°.
Assuming it is really a 90° pitch up maneuver, wouldn't the lift of the wing (though smaller due to less speed) and the missing gravity to counteract the lift make the plane go "up" (again, planes point of view!)?
Short answer: Yes. But only for a moment.
As you suggest, let's focus on the 90° case, because many things will become simpler. Intermediate climb angles can be covered by a linear superposition of horizontal and vertical flight.
Now we shoot up at 90° flight path angle (engine thrust permitting) and leave some angle of attack for the wing to create lift. Lift acts orthogonally to the local flow direction, so the lift will pull the aircraft sideways (up in the aircraft's frame of reference). Since there is no other force which will oppose this lift (drag, weight and thrust all act at 90° to the lift, parallel to the flight path), the aircraft will accelerate sideways (again, up in the aircraft's frame of reference). The angle of attack will immediately decrease and lift creation will stop — or the pilot forces the angle of attack to stay constant by pulling the elevator, and now the aircraft will further increase its flight path angle.
That is a complicated way of saying that the pilot forces the airplane into a loop. If he wants to maintain this 90° flight path angle, he better avoid creating any lift.
Answering your question, what will happen with the list depends on what maneuver the pilot is trying to do.
In the situation you are describing and taking the picture that @PeterKämpf prepared:
In your situation the gamma angle is 90º, so basically the gravity (mg) will be aligned with drag and thrust (T) will be used to compensate both weight (mg) and drag (D). Acceleration of the airplane will be T-mg-D = ma (our friend Newton). Actually, in a typical airplane, drag will be negligible compared to weight.
In this situation, there is one variable missing, the angle of attack (alpha).
Now let's go to the maneuver...
1) I would like to have my airplane stopped in the air. In this situation you try to keep the airplane at 0 speed so you don't have any lift. I have seen this done for a few seconds (or less than one) by a Typhoon in an airshow (was about 70-80º actually). It was really difficult for the pilot to keep it stable, it was moving a little bit normal to the wing.
2) You would like to have an stable climbing totally vertical. First thing is that this is not the most efficient way to do it as it is not something normal in operation. Will be the same situation as before, but there will be some speed, the pilot will try to keep the angle of attack stable to produce no lift. Problem is that such angle of attack varies with speed... so he will struggle. The Typhoon made that to scape back from the situation... he moved up and pitched down. Also climbing intercept mission in combat airplane is close to this one still very extreme.
3) You are making a loop. You create some lift to make it possible so the airplane is turning. That's better explained by @PeterKämpf
As a general comment those maneuver are usually performed in airshow by airplanes prepared for this and I have never seen any "pure" situation of what you are describing, just situations very close to what you describe. Neither combat airplanes are using it regularly although might have the capability.
However those operations are not affordable by civil airplanes. Usually thurst compensates drag and lift weight, usually lift is of the order of 10 times bigger than drag, so basically engine thrust capability is around tenth of the weight (if you consider extreme operations you will end up with a number of 2-3 times smaller). Just take the example of A320, A318 OEW is aroudn 40Tons and the thrust is 210KN (weight is twice the maximum) and going to MTWO we have 68tons (3 times). (remember the g on previous calculation)
So... normal planes usually don't have a big engine compared to their weight, however the Typhoon has a ratio of 1.15 thrust/weight.
For the plane to go straight up, there must be a force in equilibrium with the force (Lift) of the wing. That is created with the engine (Thrust) slightly off vertical in the opposite direction.
With regards to the situation where the airplane is "suspended" in a near vertical orientation. Expanding upon what Trebia Project mentioned, as well as some of the comments I read, there are tactical situations where this sort of maneuver might be used.
We went out to the area to engage two F-16's in air-to-air combat. Our flight was a flight of four A7-E's. The A7-E had great fuel efficiency, and that combined with a maximum payload of 12,000 pounds of ordinance, made it a formidable attack jet. Also, although it was not supersonic, it came close to Mach 1 at sea level. Breaking the sound barrier at these altitudes comes with a heavy cost in fuel. Anybody chasing us in burner is going to be sucking at their fuel tanks.
The specifications that made the A7-E such a good attack jet, e.g. long legs to the enemy target, were also the specifications that meant it was not particularly suited for air-to-air combat. Our rule of thumb was that if a fighter got behind your 9/3 o'clock jettison your stores and turn with your adversary. If you could last 30 seconds in the fight, the fighter will have used a good portion of their fuel. Their situation is more severe if they are in burner with you on the deck. I mention this to provide some context to the engagement with the F-16's.
When we flew attack missions, depending on the threat, we came in low or high. In either case, as a flight of 4, we were in a box formation. The lead and their wing man abreast at the head of the flight, while the second division was abreast one another a mile in trail. Four aircraft each at the corner of a square a mile a side. This mission was to help us train to maintain flight integrity while engaging multiple fighters. From the flight I learned it takes a lot of discipline.
In the area we soon found ourselves engaged with the F-16's. We were outmatched quickly and our box formation degraded into a circle with an A7 at each 90 degrees on the arc. Each A7 was protecting the six of the guy in front. A defensive position, without much chance to attack. I often took aircraft, with similar performance characteristics, vertically in a dog fight. The A7 can't maintain that geometry very long, and it becomes a game of chicken at some point. Who is willing to get to 0 airspeed with the nose at 90 degrees?
How did the F-16's handle the defensive position we took up? These guys both went vertical and sat above the circle using their burners to keep themselves "suspended." They were between 70 and 90 degrees, and looked like they were slowly falling forward when they would hit the burner and come back to a mostly stationary position. It was the coolest thing to watch. At some point one F-16 would drop in on our circle and try to get on the six of an A7. Of course the A7 in trail would force themselves around to an AIM-9L shot. With the increasing threat the F-16 would pull back up vertical, and the other would drop in on a different part of the circle. There was always one F-16 above the circle using their burner to stay atop the fight. They knew our energy was too low to get our nose much above the horizon.
Eventually the F-16 flight called the fight off with no kill. When we returned home and were debriefing the flight, the lead, who was a decorated Vietnam attack pilot, said "Yeah this use to happen in fights when I flew in Vietnam."
