There was a time before computer-based three-dimensional visualization.
There was a time before any software at all.
When James Clark Maxwell sculpted thermodynamic functions with clay.
When Louis Poinsot described the motion of the gyroscope in terms of three cones embracing each other, boiling down the essence of a free spinning top’s motion.
Today, we feed software the differential equation governing the motion of a rigid body – or talk to an AI bot soon – and produce a movie of the gyroscope’s journey. We do not need the intermediate steps of abstraction anymore.
Which means we understand the gyroscope even less, on an intuitive level.
So, I keep creating artistic tributes to Poinsot‘s work!
The Physics
The spinning top rotates quickly about its symmetry axis, and the axis itself is slowly changing with time. You can visualize it using those cones: The axis of the gyroscope rotates about the vector of angular momentum which is a constant for a body not subject to a force.
But you can see yourself standing on the spinning top – watching angular momentum rotate about you! In our usual frame of reference, angular momentum would be constant in terms of direction and magnitude. Watching it from the spinning top, its direction changes while its magnitude remains constant.
There are several ellipsoids lurking the in the physics of the gyroscope, and they help with understanding this motion. The way masses are distributed in a rigid body can be reduced to an ellipsoid. The kinetic energy of rotation can be expressed in a way reminiscent of mass times velocity squared – but now the mass is replaced by a matrix encoding that ellipsoid – Poinsot’s Ellipsoid! – and we use angular velocity. Angular momentum shows a similar resemblance to mass-times-velocity momentum.
Using these relations, the instantaneous axis of rotation can be substituted and we end up with two constraints for angular momentum in the body frame of reference: One is defined by constant energy (usually called T), one is the constant magnitude of angular momentum – L. The first one defines the surface of an ellipsoid, the second one the surface of a sphere. The axes of this ellipsoid are the inverse of the energy ellipsoid’s axes (Determined by the moments of inertia about the principal axes). The vector of angular momentum moves along the intersection curve of ellipsoid and sphere.
The Geometry
It’s one thing to write down these equations, it’s another to really feel how the intersection curves looks like. It has been a humbling experience, as usual.
I am going to cover the dance of ellipsoid and sphere in a series of geometric drawings, viewing them from different directions and varying the shape of the ellipsoid and how the sphere fits into it. In a prequel, a number 0, I am only drawing the three straight-on views – when the line of sight is parallel to one of the principal axis.
Two curved shapes are intersecting each other – how can you find the intersection? None of the usual tricks in descriptive geometry works: You cannot simply rotate a plane so that you see the plane as a line.
I think everybody has an idiosyncratic way of looking at things like that: this is how I view it! I pick a prolate ellipsoid for this visualization, a cigar. Gird the ellipsoid with a ring, around its “belly”, inside the sphere.
Now move the ring outward without distorting or bending it, in the direction perpendicular to the circle spanned by the ring. The ring becomes smaller and smaller as you move along the surface of the ellipsoid.
Now watch where this ring hits the surface of the sphere first. It has to be a pair of points in one of the planes that are perpendicular to the ring and contain the axis you move the ring along. It is a pair of points for reasons of symmetry.
If you see this “first” intersection point right in the axis in one view, you will see it on the outer edges when rotating the ellipsoid by 90°.
This is a geometrical construction based on these principles, showing an ellipsoid intersecting two spheres. The smaller sphere results in the perhaps most interesting intersection curve – when the intermediate axis of the ellipsoid just matches the radius of the sphere:
Graphite pencil construction, watercolor pencils on watercolor paper.
It is called The Plan as it looks like an engineering drawing of a spaceship.
Leave a Reply