I used to think something could rotate around more than one axis at once. Imagine a pipe sitting in space with some jets on it. Two opposing jets on the middle angled tangent to the curve firing equally would set it rotating around the long axis. Two opposing jets on the end angled perpendicularly to the pipe would set it rotating around the short axis. I thought you could do one of these and then the other and get something that was rotating around two axes at once. Then in high school I was writing some kind of space program that had objects and I needed a way to represent their rotations. Each object was fully rigid and had a position (x, y, z), a velocity (dx, dy, dz), and an orientation (ox, oy, oz), but how should I represent rotational velocity? Each one would be a vector (rx, ry, rz), but what order would I apply them in? Did that matter? How did rotational velocities around multiple axes interact? At this point I went to talk to my physics teacher, who explained that there’s no list of these velocities and when something would add a new rotation to an object it instead combines with the existing rotational velocity. Which is why gyroscopes work.
I’m curious whether I could have gotten the physics to work out if all rotation was independent, and what else would be different about that world.
I’m not sure what you mean. You’re saying it’s not possible to make a coherent mathematical description of a physics system where something rotates around multiple axes? It wouldn’t correspond to our world very well, but why are the mathematics impossible?
the angular velocity can change even in the absence of torques.
Yikes! Yes, even the model I ended up with sounds like it didn’t represent rotations properly.
Also how are you representing orientation as (ox, oy, oz)?
This was about a decade ago, so I’m not confident I remember what I did properly. But I think you can represent orientation as a one-time rotation from an initial position. So (ox, oy, oz) are a vector representing an axis with the magnitude indicating how far around that axis it rotates. Does that not work? (It’s also possible that I kept orientation as a matrix.)
Rotation is a mathematical concept, not a physical one.
In 4d, an object can rotate about two axes at once. Say the 4 coordinates are w x y z. The w and x coordinates can do the usual rotation, while the y and z coordinates rotate together, perhaps at a different rate. Or instead of 4 real coordinates, take 2 complex coordinates a and b, and have them evolve by (a,b) → (exp(i.r.t).a, exp(i.s.t).b), where t is the time and r and s are speeds.
I’m not sure what you mean. You’re saying it’s not possible to make a coherent mathematical description of a physics system where something rotates around multiple axes?
Not in 3 dimensions.
So (ox, oy, oz) are a vector representing an axis with the magnitude indicating how far around that axis it rotates. Does that not work?
I used to think something could rotate around more than one axis at once. Imagine a pipe sitting in space with some jets on it. Two opposing jets on the middle angled tangent to the curve firing equally would set it rotating around the long axis. Two opposing jets on the end angled perpendicularly to the pipe would set it rotating around the short axis. I thought you could do one of these and then the other and get something that was rotating around two axes at once. Then in high school I was writing some kind of space program that had objects and I needed a way to represent their rotations. Each object was fully rigid and had a position (x, y, z), a velocity (dx, dy, dz), and an orientation (ox, oy, oz), but how should I represent rotational velocity? Each one would be a vector (rx, ry, rz), but what order would I apply them in? Did that matter? How did rotational velocities around multiple axes interact? At this point I went to talk to my physics teacher, who explained that there’s no list of these velocities and when something would add a new rotation to an object it instead combines with the existing rotational velocity. Which is why gyroscopes work.
I’m curious whether I could have gotten the physics to work out if all rotation was independent, and what else would be different about that world.
Well, for one thing it would be mathematically incoherent.
Actually, rigid rotation is more complicated than you seem to think. While instantaneous rotational velocity (at least in 3 dimensions) is always representable by an axis and an angular velocity, the angular velocity can change even in the absence of torques.
Edit: Also how are you representing orientation as (ox, oy, oz)?
I’m not sure what you mean. You’re saying it’s not possible to make a coherent mathematical description of a physics system where something rotates around multiple axes? It wouldn’t correspond to our world very well, but why are the mathematics impossible?
Yikes! Yes, even the model I ended up with sounds like it didn’t represent rotations properly.
This was about a decade ago, so I’m not confident I remember what I did properly. But I think you can represent orientation as a one-time rotation from an initial position. So (ox, oy, oz) are a vector representing an axis with the magnitude indicating how far around that axis it rotates. Does that not work? (It’s also possible that I kept orientation as a matrix.)
Rotation is a mathematical concept, not a physical one.
In 4d, an object can rotate about two axes at once. Say the 4 coordinates are w x y z. The w and x coordinates can do the usual rotation, while the y and z coordinates rotate together, perhaps at a different rate. Or instead of 4 real coordinates, take 2 complex coordinates a and b, and have them evolve by (a,b) → (exp(i.r.t).a, exp(i.s.t).b), where t is the time and r and s are speeds.
Not in 3 dimensions.
Come to think about it, yes it can.