iiinova — where your digital self keeps evolving

Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 05/10/2026

Increasingly, I find that my guiding philosophy for living in the 21st century is the following: Nothing beautiful ever happened on a Pareto front.

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 11/09/2026

Thinking about the dedication to Misner, Thorne and Wheeler's "Gravitation" today. Succinct, beautiful, and demonstrates the caliber and perspective of the writing that follows it.

The dedication to the textbook Gravitation by Misner, Thorne, and Wheeler. It reads as follows:

"We dedicate this book
To our fellow citizens
Who, for love of truth,
Take from their own wants
By taxes and gifts,
And now and then send forth
One of themselves
As dedicated servant,
To forward the search
Into the mysteries and marvelous simplicities
Of this strange and beautiful Universe,
Our home."

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 09/06/2026

Another GIF on the archives from one of my favorite mathematical facts: Taylor series are a special case of (generalized) Fourier series! A Taylor series is the generalized Fourier series, using a monomial basis, of a function in the limit where the approximation/integral range becomes a point.

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 23/05/2026

An old classic from my archives: Two simple pendula swing from a base that slides with friction along a horizontal rail. The friction eventually causes the base to stop sliding, and the pendula to anti-synchronize! A simple, cool example of spontaneous synchronization.

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 18/05/2026

An unfinished snippet from my dynamics notes; I’m framing the question of whether or not non-conservative forces truly exist as a debate between two entities. Hoping I can polish their “design” a bit before finishing my notes!

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 14/05/2026

How to get singularity-free equations, then? My approach; parametrize the position using the /cosines/ of the angles directly, rather than the angles themselves. The equations are gnarly (they're going to be no matter what), but they are both singularity- and trigonometry-free!

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 14/05/2026

You may think that the singularities are caused by gimbal lock-type parametrization problems. But in fact, adding an extra angle (and an appropriate constraint) to avoid that actually makes it WORSE!

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 14/05/2026

Why is it tricky? Well, if you use the typical approach--- the Lagrangian method, parametrizing rod orientations with Euler angles---your equations of motion become singular whenever either rod hits a vertical orientation!

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 14/05/2026

One of my students' tasks for their final dynamics assignment; simulating a 3-D double pendulum! Here it is, started from rest, along with the "shadows" it casts on each Cartesian coordinate plane. Simulating it properly is surprisingly subtle!

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 13/05/2026

This is one reason internships are critical, not just for students, but for universities; giving students early concrete proof that skills they're developing actually matter for their future jobs is an excellent motivator for them to actually care about mastering those skills!

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 13/05/2026

The effect on study time is obvious. But another factor is students seeing a divide between the kinds of jobs they can get as students (service industry jobs, usually) and what they're studying, and seeing a disconnect between what gets them money and what they're training for.

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 13/05/2026

In my opinion, a heavily overlooked factor in declining college student performance over the last decade is the growing number of students that are forced to work while enrolled.

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 11/05/2026

It is impossible for me to know what happens next; in two weeks, in a year, etcetera. But the sun will still rise, and people will still dream, and will still need helping reaching those dreams. So I’ll keep at this professor thing, for as many sunrises as this dream has got left.

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 11/05/2026

It may well be that my dream career as a professor gets cut short by this change. But I’ve already had most of my dreams come true; and what has given me the most satisfaction as a professor has been helping students get jobs and reach places where they can pursue /their/ dreams.

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 11/05/2026

Is the world I grew up in gone? It seems so, and I mourn it. But a new world is forming for our youth, and they have the right to mold it into whatever the shape of their dreams is. All I can do is what any teacher hopes to do for their students—all I can do is help.

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 11/05/2026

But the world is changing, and the old world is giving way to a new one. All I can do is try to convince my students that they can have a part in building it—that the structures of power that exist now do not have to dictate what their new world will be like. That they, ultimately, matter.

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 11/05/2026

The world seems simultaneously rigid and crumbling; covered in systems and structures that impose on every facet of life, but which feel pointless and/or harmful. Everything feels rent-seeking; like these systems are the “price” for participating in society, while none of them actually help anyone.

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 11/05/2026

Another semester nears a close. Thinking of my students, my biggest concern isn’t AI per se—it’s how to help them be independent, leaders, and agents of change in a world that seems to desperately want them to be none of these.

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 28/03/2026

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!

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 28/03/2026

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".

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 28/03/2026

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!

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 18/03/2026

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.

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 11/02/2026

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)

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 11/02/2026

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).

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 11/02/2026

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

aghostinthefigures.com

The Notion of Motion

Lecture notes for my analytical dynamics course in the University at Buffalo.

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 06/02/2026

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.

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 05/01/2026

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!)

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 23/12/2025

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

aghostinthefigures.com

Part VIII: Navier-Stokes, Existence & Uniqueness

What does the Navier-Stokes equation represent? Why is it so difficult to deal with, and what are the philosophical implications of that?

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 23/12/2025

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!

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 26/11/2025

A preview of a context-less, interesting plot coming out of my research. It has applications, I promise!

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 10/10/2025

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!)

lathisms.org

Lathisms

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 07/08/2025

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.

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 07/08/2025

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.

youtu.be

Lecture 33: Thermoregulation and extreme heat events

YouTube video by Arnaldo Rodriguez-Gonzalez

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 07/08/2025

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.

youtube.com

Commentary on Heat Transfer - YouTube

These are a series of lectures designed for the summer undergraduate heat transfer course in the University at Buffalo's Mechanical and Aerospace Engineering...

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 21/07/2025

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...

youtube.com

Commentary on Heat Transfer - YouTube

These are a series of lectures designed for the summer undergraduate heat transfer course in the University at Buffalo's Mechanical and Aerospace Engineering...

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 21/07/2025

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.

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 21/07/2025

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!

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 21/07/2025

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.

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 21/07/2025

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.

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 21/07/2025

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.

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 21/07/2025

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.

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 21/07/2025

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.

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 21/07/2025

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?

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 21/07/2025

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.)

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 21/07/2025

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.

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 21/07/2025

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!

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 21/07/2025

One of my favorites parts of teaching heat transfer: Explaining that radiation is not /technically/ a form of heat transfer! (A thread.) #physics

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 20/06/2025

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!

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 19/06/2025

Anyways, this is a relatively self-centered overshare, which I’ll likely delete later (too risky to share things on social media now), but I’m not alone in feeling this way. I guess this is just my way of grieving what was, and what could’ve been. It also explains why I post so rarely now.

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Arnaldo Rodriguez-Gonzalez @aghostinthefigures.bsky.social · 19/06/2025

Although it seems like this’d be good on the “consumer” side—higher quality for far cheaper on written content, for example—my guess is that this’ll heavily discourage potential authors from even considering writing in the first place. We will go the same way that stonemasons did.

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