# Why is modeling a physical system with ODEs sufficient?

I've read a few papers in dynamical systems where the model equations are sets of ODEs, with the state space, say, the spatial variables x, y, z, and an angle variable phi all evolving forward in time. The simulations match the 3D lab experiments very well too.

Why does this happen?

I would think that 3D phenomenon needs to be modeled with PDEs, using partial derivatives to describe, say, a change in x per change in y.

How do ODEs make good model equations?

Seems weird...

• You need to specify if you are considering the mechanics of solid bodies or the physics of some medium. Perhaps cite the examples/papers you have in mind more specifically. There is some intermediate stage of multi-particle simulations, for instance asteroid belts or planetary rings can, to some approximation, simulated as continuous mediums, I do not know if more like a fluid or a gas. – Lutz Lehmann Nov 29 '19 at 12:08
• So you have the opportunity to simulate the aerodynamical flow around that object via Navier-Stokes PDE or just add a drag term to the force balance with a shape dependent drag coefficient. Both are routinely done, and give good approximations to the physical reality as long as the conditions used in establishing the simulations apply (non-turbulent flow, sub-sonic speeds etc.) – Lutz Lehmann Nov 29 '19 at 18:30

## 1 Answer

Every model is only as good as the approximations that are made in deriving it. Sometimes the approximations that reduce a PDE model to an ODE model are so good that the resulting ODE model is accurate enough to describe everything we want to know about the object.

Here's an example: Think about a spacecraft traveling through the solar system. A complete model would take into account the flexing of the structure, the vibrations of the body, its thermal response to heating from the sun, and many other factors. But if we just want to know where the spacecraft is going to be at a given time, then all of these effects are so small that they can be neglected and all that matters is (i) how Newton's law of gravity acts on the center of mass, and (ii) what direction the antenna points to. This can be described with an second-order ODE system with 5 components (3 positions + 2 direction angles), rather than a PDE system. The reduced system is accurate enough to predict the position of a space craft flying to Mars to within a few hundred meters. It is of course true that at least in principle a PDE model could be more accurate -- but there are so many things about the ODE model we already don't know accurately enough (e.g., the initial position and velocity) whose combined resulting uncertainties are larger than the difference between the ODE and PDEs models that there is really no point using the PDE model: The resulting model will not be more accurate because we still don't know initial positions and velocities accurately enough.

• I think that depends on the problem you're trying to model. Sometimes the third dimension is just like the first two (e.g., if you're trying to model the trajectory of a particle in an external force field), and sometimes it's not (e.g., the third variable could correspond to an internal "bending moment" of a body that moves in a 2d plane. – Wolfgang Bangerth Nov 30 '19 at 18:27
• @user33455: The state of a rigid three-dimensional body that has no internal degrees of freedom is really just characterized by 5 variables -- its center of mass, and the angles of its orientation in space (e.g., the angles the tip of the antenna makes against the positive z- and x-axes of a coordinate system that goes through the center of mass). – Wolfgang Bangerth Dec 16 '19 at 19:22
• Well, first, if you have a model that matches experiments, why would you want to make it more complicated? Complex models are expensive to validate without, apparently, providing any closer match to reality. But if you really wanted to make your model more complicated, then yes allow for a third axis and rotations in that direction as well. – Wolfgang Bangerth Dec 17 '19 at 0:06