Eule Slick

A couple months ago, I talked about the LIGO work I'm doing here in Florida, but I'm also working on the LISA project, a gravitational wave detector in space. The detector consists of 3 spacecraft that orbit the Sun trailing slightly behind the Earth:

Wikipedia

Recently I've been developing a simulation of the LISA spacecraft, specifically their orientation to face each other. This gets into a part of Physics that I don't have a lot of experience with: rigid-body dynamics.

Most of the time, physicist can get away with considering things to be points or spheres, but things get more complicated with asymmetrical objects. Suppose we have a box that we want to rotate around two axes, x and y. Depending on which we do first, we get different results:

To get around this ambiguity, the mathematician Leonhard Euler (pronounced "oiler") realized you can specify the orientation of a 3D object by defining 3 rotations from a principle set of axes:

Wikipedia

I won't get into the details of how the angles are defined, but the result explains some really interesting effects. Here's a video recorded by a NASA astronaut aboard the space station of a "T-handle" spinning:

That strange tumbling behavior is explained by Euler's rotation equations:

where ω is the angular velocity, and τ is the torque applied. The I is the moment of inertia tensor, which requires some explanation: Simply put, it's the rotational equivalent of mass, which quantifies how a torque relates to an angular acceleration. What complicates things is that I depends on your choice of axes. It's a simple diagonal matrix in the frame where the T-handle is fixed, but that frame rotates. The fact we're working in a rotating frame results in the second term in the equation above.

Here's where the Euler angles come in: The differential equation above tells us how the angular velocity is changing, but we're interested in how the orientation of the handle changes. We need to relate the angular velocity to the change in the Euler angles, and include those in our integration. Putting everything together, we can simulate how the handle reacts to an initial rotation.

First with the spin perfectly aligned, nothing too interesting happens:

If we introduce just a 2% offset in the angle though...

we see precisely the sort of flip from the video! I've said it before here, but this is why I love Physics: With just a few relatively simple equations, you can explain even the weirder parts of reality.