And here's rotation around the "long axis".
You can see all this in real life by flipping a book/smartphone into the air along each of its "principal axes"; you'll be able to replicate the Dzhanibekov effect easily. Just make sure your phone has a case so you don't break it while tossing it around!
This doesn't happen for rotations around the "short" and "long" axes of the object; small perturbations in the rotation remain small, and only translate into tiny pole wobbling. No flips!
Here's rotation around the "short axis".
Tackled animating rigid-body motion for my analytical dynamics course; take a look at the Dzhanibekov effect!
For generic 3-D objects, rotations around the "intermediate axis" are unstable, and lead to periodic pole flips!
Every single time I try to calculate anything involving 3-D rigid body dynamics, I can't help but feel like someone's just slightly out of view, pointing and laughing.
Attitude dynamics, when analyzed properly, is so complex! It feels like angles are some kind of eldritch abomination.
Ultimately, I'm doing it because it's the only way I would've written something like this—my own perspective on the subject of dynamics—and writing something is ultimately better than writing nothing. Not elegant reasoning, admittedly, but it counts.
Anyways, to whomever reads these; enjoy. (3/3)
Perhaps it's a kneejerk reaction to AI slopwriting (although I'm not naive enough to think AI can't fake handwriting). Maybe it's because I'm fond of hand-writing; one of my favorite hobbies is writing letters. Maybe it's because it's easier for me to do dynamical diagrams/notation by hand. (2/3).
I've been making lecture notes for my analytical dynamics course and uploading them to my site. They're hand-written, but can be read standalone.
aghostinthefigures.com/the-notion-o...
I wrote 100 pages and haven't even gotten to dynamics yet. Why on Earth am I doing this? (1/3) #physics #dynamics
Adventures in teaching graduate-level dynamics:
I have done the equivalent of asking my students to "fetch the blinker fluid"—asking for a coordinate-free representation of the "direction cosine tensor". Truly diabolical.
Obligatory conference post: I’ll be at #JMM2026 presenting some of my upcoming work on predicting the existence of qualitatively periodic orbits in dynamically stabilized systems.
(It’s my first invited talk at a conference; wish me luck!)
If you’re curious to know what all the hubbub on Navier-Stokes is about and why the equations are so problematic, I briefly wrote about it in my Commentary on Fluid Mechanics:
aghostinthefigures.com/2019/10/28/p...
#physics #math
Curious (and a bit baffled) to see all the hype surrounding the Navier-Stokes equation in the context of the Millenium Prize these days; from my perspective, the Millenium Prize N-S problem is about the least interesting parts of what we don’t know regarding Navier-Stokes!
A preview of a context-less, interesting plot coming out of my research. It has applications, I promise!
Honored to have been featured as today’s mathematician in Lathisms’ annual calendar for Hispanic Heritage Month!
(Also delighted to see some familiar faces in this year’s cohort!)
I hesitate to share things publicly these days, for perhaps obvious reasons. But I suppose that the fear of evil should not drive inaction; the hope for good should drive action instead. And so I have to proverbially put my money where my mouth is.
I hope they're useful to someone out there.
If anything, thinking of making this lecture on thermoregulation and extreme heat events motivated me to make and post these recordings in the first place; although highly simplified, it is important knowledge for anyone invested in the global health of humanity this century.
I've uploaded my lectures for my summer heat transfer course on YouTube, free to watch.
They are largely unedited and a tad sloppy; but I made a commitment years ago to the free distribution of knowledge, and I have to see it through.
That's why radiation is not technically a transference of heat!
If you'd like to learn more about heat transfer, I've made my (currently incomplete) summer course lectures on heat transfer available for free on YouTube: check them out if you like!
www.youtube.com/playlist?lis...
And, key to note, no actual heat flow occurs in radiative temperature equilibration. /Electromagnetic/ energy flows between the objects, /some/ of which is converted back into thermal energy; and since the flow is affected by the objects's temperatures, this (potentially) leads to equilibrium.
This means that modeling radiation involves modeling three separate physical processes; the emission of electromagnetic energy, its propagation through space, and its (partial) conversion back into thermal energy upon interacting with matter.
THIS is what makes modeling radiation so hard!
What happens, in short, is that objects will emit /electromagnetic/ energy into their surroundings (in the form of light) as a function of their temperature. That electromagnetic energy travels through space, and upon interacting with matter, may be partially converted back into thermal energy.
If the temperature of something is based on the random kinetic energy of its molecules, what is the temperature of empty space? If empty space has no temperature, how can it have thermal energy, and therefore how can heat flow occur in it?
The answer; it doesn't! No heat flow occurs in radiation.
The field of study called "heat transfer" is then called this because most of its modeling efforts are focused on relating heat fluxes and heat transfer rates to temperature imbalances, as well as other properties of the physical system being studied.
But this idea breaks down for radiation.
Heat flow then occurs as a result of imbalances in temperature, like how fluids flow under pressure imbalances! A temperature imbalance in matter causes a flow of thermal energy from hot to cold that "corrects" the imbalance over time, in the absence of "thermal forcing" or anything like that.
For the first two modes, we hypothesize that there is a quantity—thermal energy, or heat—that is a function of the temperature of the object/point being studied, which can flow through matter and be transferred between objects.
However, we observe that temperature equilibration can occur even between two objects separated by empty space, with no matter between them! This is the mode we call "radiation".
So this /empirically/ describes the three modes of "heat transfer". How do we model and describe them?
Between solids in contact (and within them), we call the mode that induces temperature equilibration "conduction".
Between solids and fluids in contact, or between just fluids, we call that mode "convection". (There's a bit more refinement to these descriptions, but this describes most of it.)
What we call "modes" of heat transfer (in the field-of-study sense) are just different physical contexts in which we observe a key empirical behavior; that objects with different temperatures, in the absence of "thermal perturbations", will eventually "equilibrate" to the same temperature over time.
To explain why, we first need to answer what the /field of study/ we call "heat transfer" is.
In short, it's the field of study that seeks to describe how temperature changes, in time and in space, across and within objects. Notice no actual definition of what "heat" is!
One of my favorites parts of teaching heat transfer:
Explaining that radiation is not /technically/ a form of heat transfer! (A thread.) #physics
I saw this model work well for artists and writers I know; it comes with its own challenges (who owns IP/usage permissions, collective identity, interpersonal disputes, etc.) but I agree that this might be a good way forward for folks starting out.
Community always helps!
