Representation in the Glider model
Applied mathematical modelling (and theory more generally) is a mysterious thing – and one of my favorite things to ponder. Once we have a good model in-hand, all sorts of possibilities are open before us, thanks to calculation. But arriving at a good model is black-magic. To bastardize an analogy from Carl Sagan, a model is a bit like a candle in the dark – a spot of illumination. All around that model is chaos and darkness, and occasionally a monster lurking. As long as we stay close enough to that candle/model and don’t let the facts wonder too far from the model hypotheses, we are safe. But sometimes our models are wrong. Sometimes we wonder a little too far without realizing it, still assuming that we’re in the halo of the known. And that’s when the monsters get us. Castle Bravo, Space Shuttle Columbia, Lion Air Flight 610, Tacoma Narrows Bridge,…
Models are sharply defined things – they are ALWAYS built on hypotheses, and they can NEVER escape the limitaitons of the hypotheses they are built on. Without hypotheses, a model can not exist, but hypotheses are always wrong in some way. So, a good modeller is always worrying about the monsters lurking in the dark around their model. Today, I gained a little more insight into how monsters hide in models. The big-picture take-away is that representation matters, and its good to explore alternative representations.
I was pondering Lanchester’s classic glider model, which I was teaching in class as an example application of Newton’s laws. It is a beautiful model – simple, but with clear physical motivation, and it’s analysis leads to fundamental insight about the basic properties of aircraft flight – properties that are only intuitive in hind-sight. The classical formulation of Lanchester’s glider model appears through Newton’s laws by adding a lift force to the equations for projective flight: \[\begin{gather} m \ddot{p} = F_{\text{gravity}} + F_{\text{drag}} + F_{\text{lift}}. \end{gather}\] This eventually leads to a pair of equations for the speed \(v(t)\) and the pitch \(\gamma(t)\) of the glider, as \[\begin{align} \dot{v} &= - \sin \gamma - \delta v^2, \\ \dot{\gamma} &= -\frac{ \cos \gamma}{v} + v, \end{align}\] where the lone parameter \(\delta\) is the ratio of the drag coefficient relative to the lift coefficient. We can then analyze these equations for different values of \(\delta\), and find that when drag dominates, the behavior is projectile-like, but when drag is small, we predict something resembling level flight. The analysis is all very fun, and leaves one with a bit of satisfaction and confidence after comparison with some videos of simple gliders.
But there is a slightly different way to motivate Lanchester’s glider model. It leads to equivalent equations, but these equations FEEL different. They lead to questions that aren’t even in-frame in this first formulation, and possibly reveal how close the darkness is on one side of the model.
The key starting point is an intermediate point in the derivation of the glider equations. After making the forces in Newton’s law explicit, we reach a pair of equations for speed \(v\) and pitch \(\gamma\) of the form \[\begin{align} m \dot{v} &= - m g \sin \gamma - C _ D v^2, \\ m v \dot{\gamma} &= - m g \cos \gamma + C _ L v^2. \end{align}\] where \(C _ D\) is the coefficient of drag and \(C _ L\) is the coefficient of lift. One might notice here that they are both multiplied by the squared speed \(v^2\) – just that one part is serving as a drag term while the other is serving as a lift term. But in both cases, it is the forward motion of the glider interacting with the air to create forces.
It seems, then that maybe instead of thinking of lift and drag as separate forces, we should instead be thinking of them as components of the same force, \[\begin{gather} F_{\text{air}} := F_{\text{drag}} + F_{\text{lift}} = v^2 \begin{bmatrix} C _ D \\ C _ L \end{bmatrix} \end{gather}\] and that it is really just this force that is dependent on the speed of the glider, with names assigned to components in particular directions relative to the plane.
Well, this all makes sense. Drag and lift are two parts of air’s tendancy to resist the motion of the glider, and it is as much about the direction that force is exherted in, as much as the magnitude of that force. So, it make sense to reparameterize this in terms of the direction and magnitude as \[\begin{gather} F_{\text{air}} := F_{\text{drag}} + F_{\text{lift}} = C_A v^2 \begin{bmatrix} \cos q \\ \sin q \end{bmatrix} \end{gather}\] where the direction \(q\) is going to be controlled by the shape and orientation of the glider.
Now, the gloom around our model recedes! While air-resistance is always present, exactly how it manifests depends on the object we are considering.
While for cannonballs and symmetric projectiles, it’s easy to surmize that the force of resistance is anti-parallel to the direction of motion. But general falling objects, it seems quite likely that air resistance will not act purely anti-parallel! Thus, lift force is not really a special thing, but a generic thing that may act in any direction orthoganal to the direction of motion of an irregular falling body.
If this falling object is rotating chaotically as it falls, the direction and magnitutde of the force of air-resistance will change as it turns. But if, like a glider, the orientation tracks the direction of movement, then it is reasonable to say the directions of drag and lift are approximately constant relative to body’s orientation, and Lanchester’s model applies.
But we really never closely considered the importance of glider orientation in the naive derivation of Lanchester from projectile motion; those assumptions were backed in way back when we started imagining the trajectories of cannonballs, and the assumption of \(\delta\) constant seemed to need no extra justification. Now, we see there are allot of other things going into \(q\) that could potentially important – the monster hiding places have been exposed. It’s not even a very large step to suppose \(q\) itself is a dynamic quantity needing it’s own governing equations, and to wonder what conditions are necessary for its stable solution.