The Phugoid model is a system of two nonlinear differential equations in a frame of reference relative to the plane. Let v(t) be the speed the plane is moving forward at time t, and (t) be the angle the nose makes with the horizontal. As is common, we will suppress the functional notation and just write v when we mean v(t), but it is important to remember that v and are functions of time.
If we apply Newton's second law of motion (force = mass × acceleration) and examine the major forces acting on the plane, we see easily the force acting in the forward direction of the plane is
This matches with our intuition: When is negative, the nose is pointing down and the plane will accelerate due to gravity. When > 0, the plane must fight against gravity.
In the normal direction, we have centripetal force, which is often expressed as mv2/r, where r is the instantaneous radius of curvature. After noticing that that = v/r, this can be expressed as v, giving
Experiments show that both drag and lift are proportional to v2, and we can choose our units to absorb most of the constants. Thus, the equations simplify to the system
It is also common to use the notation for and for . We will use these notations interchangeably.