I want to use the following equations so as to describe the motion of parafoil $$\left[ {\begin{array}{*{20}{c}}{\dot x}\\{\dot y}\\{\dot z}\end{array}} \right] = {V_a}\left[ {\begin{array}{*{20}{c}}{\cos ({\gamma _a})cos({\chi _a})}\\{\cos ({\gamma _a})sin({\chi _a})}\\{ - sin({\gamma _a})}\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}{{w_x}}\\{{w_y}}\\{{w_z}}\end{array}} \right], $$ where ${\gamma _a}$ is a flight path angle, and ${\chi _a}$ is heading angle in horizontal plane.

$${\gamma _a} = - arcsin \frac{{{V_{az}}}}{{{V_a}}}$$ and also

$${\gamma _a} = arctan \frac{{{-1}}}{{{L/D}}}$$ The equations for lift force and drag force lool like these

$${\displaystyle L\,=\,{\tfrac {1}{2}}\,\rho \,V_{a}^{2}\,C_{L}\,A}$$ $${\displaystyle D\,=\,{\tfrac {1}{2}}\,\rho \,V_{a}^{2}\,C_{D}\,A}$$

according to paper https://lib.dr.iastate.edu/cgi/viewcontent.cgi?referer=https://www.google.ru/&httpsredir=1&article=1629&context=etd

But why the reference area $A$ and speed is similar for both cases? I think its projection should be rather smaller for drag force than for lift one.

But then on can make assumption that $${\gamma _a} = arctan \frac{{{-1}}}{{{C_{L}/C_{D}}}}$$ I don't understand why this is valid for parafoil

Anyway, I don't want to sink in rigid body physics and it will be comfortable for me to stay on next level of abstraction.

For my case the maximun lift-to-drag ration $L/D = 4$ and the only information i know is the area of parafoil. It's enough to characterize the simple model but i can't describe the change of speed caused by the change of lift-to-drag ratio due-to symmetric motion of control handles (breakes, srtopes? I'm not sure what defenition is correct). What is the equation that is responsible for the dependence of lift-to-drad ratio on symmetric motion of handles? Or is there other way to write down variation of velocty due-to handles lengths that allows not to consider forces? It would be appropriate if you know how it was done in X-38 parafoil.

In my opinion the shoud be the law like this $${\gamma _a} = {\mathop{\rm artan}\nolimits} \left( {\frac{{ - 1}}{{L/D}}} \right) = f(\delta )$$ where $\delta$ is symmetric lenghth of each of two control handles. But as I said I have only area of parafoil that is 400 ${m^2}$ and mass a few ton.