I assume you are going to program this. First you need to write the vector differential equations of motion. The derivative of position is velocity, and the derivative of velocity is acceleration. Acceleration is force divided by mass. Force is the sum of a number of components, mainly gravity, lift, drag, and thrust.

Then you just integrate those with any ODE solver you like. The simplest is Euler. If you want more accuracy, you can use Runge-Kutta. You probably don't need a stiff solver such as Gear or Adams-Moulton.

So if the motion of the aircraft is along a straight line with wings level at constant speed, then the lift vector is equal and opposite to the gravity force vector. The way you turn it is to bank the wings to an angle. For example, if you bank the wings 45 degrees to the left, that tilts the lift vector 45 degrees to the left, so it's vertical component is .707 of what it was, and .707 of it is directed to the left. The sideways force causes the path to be a circular arc. The reduced vertical force causes the aircraft to start descending.

To compensate for that tendency to descend, the pilot increases the magnitude of the lift vector by adding back pressure on the control yoke. The increased lift results in increased drag, so the pilot increases engine power to maintain speed.

Then when the aircraft has travelled far enough around the circular arc to be headed in the desired direction, the pilot levels the wings, releases the back pressue, and reduces the engine power. If he doesn't, the aircraft will start to climb.

You'll notice this the next time you fly.

That's how turning works. Now I'll tell you how straight flight works. The lift and drag you get from the wing depends on two things, speed and angle of attack. If you reduce speed, but want to stay in level flight, you need more angle of attack to get the same lift. So the way a pilot slows down a plane is to reduce power and then gradually pull the nose up by applying back pressure on the yoke.

Since it would be tiring to continue to apply back pressure when flying slowly, the pilot has a "third hand", the trim wheel. Rolling that wheel backward applies back pressure on the yoke so the pilot doesn't have to hold it. In fact, the primary speed control of the airplane is the trim wheel. The power doesn't actually control speed, it controls whether the aircraft ascends or descends at the speed it is travelling.

One last point is balance. If you take a plane on the ground, and place a jack under each wing at the center of lift and hoist it up, its nose will fall toward the ground. It's center of gravity is forward of the center of lift. So when it's flying, as the main wing is pushing up, the tail is pushing down. The moment between the two is what keeps the nose from dropping. This is very important for safety, because it stabilizes the speed. If the speed decreases, that moment decreases, and the nose drops, causing the aircraft to start to go "downhill". Since it is going downhill, it picks up speed. Conversely if it's speed increases, it starts to go uphill, causing its speed to decrease. If the aircraft is loaded too far aft, its speed and its up/down directional stability is lost. In fact, it is possible to get nosed up, and then go into a backslide all the way to the ground!

I assume you are going to program this. First you need to write the vector differential equations of motion. The derivative of position is velocity, and the derivative of velocity is acceleration. Acceleration is force divided by mass. Force is the sum of a number of components, mainly gravity, lift, drag, and thrust.

Then you just integrate those with any ODE solver you like. The simplest is Euler. If you want more accuracy, you can use Runge-Kutta. You probably don't need a stiff solver such as Gear or Adams-Moulton.

So if the motion of the aircraft is along a straight line with wings level at constant speed, then the lift vector is equal and opposite to the gravity force vector. The way you turn it is to bank the wings to an angle. For example, if you bank the wings 45 degrees to the left, that tilts the lift vector 45 degrees to the left, so it's vertical component is .707 of what it was, and .707 of it is directed to the left. The sideways force causes the path to be a circular arc. The reduced vertical force causes the aircraft to start descending.

To compensate for that tendency to descend, the pilot increases the magnitude of the lift vector by adding back pressure on the control yoke. You'll notice that you feel a little heavier in the turn. The increased lift results in increased drag, so the pilot increases engine power to maintain speed.

Then when the aircraft has travelled far enough around the circular arc to be headed in the desired direction, the pilot levels the wings, releases the back pressue, and reduces the engine power. If he doesn't, the aircraft will start to climb.

You'll notice this the next time you fly.

That's how turning works. Now I'll tell you how straight flight works. The lift and drag you get from the wing depends on two things, speed and angle of attack. If you reduce speed, but want to stay in level flight, you need more angle of attack to get the same lift. So the way a pilot slows down a plane is to reduce power and then gradually pull the nose up by applying back pressure on the yoke.

Since it would be tiring to continue to apply back pressure when flying slowly, the pilot has a "third hand", the trim wheel. Rolling that wheel backward applies back pressure on the yoke so the pilot doesn't have to hold it. In fact, the primary speed control of the airplane is the trim wheel. The power doesn't actually control speed, it controls whether the aircraft ascends or descends at the speed it is travelling.

One last point is balance. If you take a plane on the ground, and place a jack under each wing at the center of lift and hoist it up, its nose will fall toward the ground. It's center of gravity is forward of the center of lift. So when it's flying, as the main wing is pushing up, the tail is pushing down. The moment between the two is what keeps the nose from dropping. This is very important for safety, because it stabilizes the speed. If the speed decreases, that moment decreases, and the nose drops, causing the aircraft to start to go "downhill". Since it is going downhill, it picks up speed. Conversely if it's speed increases, it starts to go uphill, causing its speed to decrease. If the aircraft is loaded too far aft, its speed and its up/down directional stability is lost. In fact, it is possible to get nosed up, and then go into a backslide all the way to the ground!