The white room looked as featureless and as plain as he remembered. Peeking out from behind the door, the individual explained, “So I got really busy.”

“Where did you go?” Theodore questioned.

“I went to the woods.”

“The woods? What for?”

“To suck out all of the marrow in life.”

Theodore gave the individual a funny look.

“Henry David Thoreau? I went to the woods because I wished to live deliberately? Okay, I went foraging for raspberries.”

“Raspberries? What are we doing today?”

“I was wondering if we can try to go over the Euler-Lagrange equation.”

Theodore spat out a mouthful of water. He felt the cold glass of a half-filled cup in his hand. Since when did he have a glass of water in the first place? “What?”

“I want to know where the Euler-Lagrange equation came from. Like why they made it.”

Theodore felt beads of sweat forming on the sides of his forehead. Could he answer the individual’s inquiry as comprehensively as he desired? Was the individual testing him? He squirmed. Something felt out of place. “Why are you asking me this?”

“What do you mean? You told me about it before as an alternate way of solving classical mechanics problems.”

“I did?”

The individual impatiently reminded Theodore, “It’s easy to mistake those we do not expect.”

“What?”

The individual gesticulated down their body. Theodore shook his head in confusion.

“Can you look closer? It’s me. Not the other one.”

Theodore leaned in. The individual who stood before him was not the one who he had painted and swam with. Rather, they were the one who had asked him about the hoop and beads from a couple of weeks ago.

Seeing Theodore understand, the individual flicked on a mini flashlight, pointing the light at an angle into the water of Theodore’s cup. One could perceive the ray of light bending. The individual continued, “I know the Euler-Lagrange equation has something to do with Fermat’s Principle of Least Time. Light always takes the path with the shortest time between two points. That’s why it bends at the water-air interface.”

As if the tarnished cogs of the gears in his mind creaked into commission, he remembered teaching a student about these things. With all of it coming back to him, Theodore coughed, regaining his composure. Swiping off the sweat on his forehead with his forearm, he procured a brush and a bucket full of black paint. Dipping the brush tip into the inky black, he dotted two points on the milky white ground, labeling them points A and B.

“Suppose you have a place where you want to begin and a place where you want to end. Point A can be our starting point and point B can be our endpoint. You might naturally ask how we will get from A to B?” Theodore proceeded to draw a straight line between the two points. “Maybe we can go straight?” Then, he drew another path, much more convoluted with arbitrary bends and curves. “Or maybe like this?” Theodore continued sketching out more and more paths, stating, “The fact of the matter is there are many ways to get from one point to another without any rules in place. There are indeed infinitely many possibilities.”

The individual reasoned, “So why does the way light travels from one point to another in a medium go in a straight path rather than some weird path? Is it because there’s some rule in place?”

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“Yes! That rule is Fermat’s Principle of Least Time that you just mentioned. It is a suitable analogy to what we’re about to do here.” Theodore transmuted his glass of water into a baseball. “Suppose right here in my hands, where the baseball currently rests, is point A. I will now throw the ball towards an intended target, point B, on the wall over there-ish.”

Some twenty feet away, a standard target board protruded out from the wall. Theodore dunked the baseball in the black paint, throwing it. As the ball arched in its path, the black paint that trailed behind it permanently stained the air. Splat! The ball gravely missed the target board, woefully rolling away on the floor like a teenager rejected for prom. The individual laughed.

Theodore pointed at the black arch of paint. “What do you call the shape of this curve?”

“It’s a parabola.”

“Yes. Just like the light going from point A to point B, we see the baseball taking a specific type of path when I throw it. In this case, rather than a straight line, it’s a parabola. So what’s the rule?”

The individual contemplated. “Is it gravity?”

“Gravity! Gravity makes things we throw on Earth rise and fall in parabolas.”

“So what does the Euler-Lagrange equation have to do with all of this?”

“Usually when we want to figure out what path something takes, we use F=ma, Newton’s 2nd Law. We look at an object. We ask what forces are acting on it. Then, we can see how those forces influence how the object moves. So the force of gravity can make an object move in a parabola. The electric force can make light bend when it goes from one medium to another.

We continue to do this at every point, using F=ma, updating along the way. It’s basically cause and effect. For instance, if I slapped you (the cause), you’d slap me back twice as hard (the effect).”

“Okay.”

“The Euler-Lagrange equations use a different approach. They assume nature prefers the most efficient route. If given the rules of the universe, nature will work its way around it such that the path it creates is one of least effort. The mathematical process of this is called the minimization of the action through a defined quantity called the Lagrangian. That’s what Wikipedia will probably tell you. The important thing is, instead of caring about cause and effect, the Euler-Lagrange equation insists we only focus on the beginning, the end, and the rules of the universe. With these three truths, the Euler-Lagrange equation can determine the shape of the path for any object. In other words, it has all been predetermined in a sense.”

“How is it predetermined?”

“It means, well, going back to the slapping example, if we use the Euler-Lagrange perspective, we can say you would have slapped me twice as hard not because I slapped you first, but because the universe thought this ought to have been the path of least effort. You could have blowtorched, electrocuted, waterboarded, or tortured me however you pleased. But no. You slapped me, and the universe wouldn’t have had it any other way. That’s how it must have always been.”

“Then, doesn’t that mean I don’t get to make my own choices since everything is already set in stone?”

“The Euler-Lagrange equation only provides a perspective on how our universe functions. Perhaps you still have free will. Or perhaps everything is, indeed, set in stone. Interestingly enough, you can use the Euler-Lagrange equations to derive F=ma and vice versa. So it’s really whatever way you want to look at things.”

The individual nodded, quite gratified with Theodore’s last statement. They acknowledged, “I see. But since either way works, then, why do we use Euler-Lagrange if we can always use F=ma?”

“Oh, because it’s useful,” Theodore dryly answered.

“Very helpful.”

“Because there are many classical mechanics problems out there that are impossible to solve with F=ma.”

“Can you show me?”

“Yeah! Let’s do an example, something a bit more spicy than the projectile motion of a baseball but not too mathematically tedious. How about the simple pendulum?”

“Sure. I know that one.”

Theodore eagerly dipped his paintbrush into the paint bucket. The individual copied him with their own brush.

Working together, nitty-gritty lines of work involving the time derivatives of the cartesian components and the partial derivatives in respect to both the position and velocity gradually populated the walls. In the process, Theodore and the individual conversed along each step of the way.

“You see here, we now apply small-angle approximations so that we can use a Taylor series expansion to reduce this into a homogeneous second order differential equation,” Theodore expounded.

“Can we solve it using separation of variables?”

“It turns out you can by knowing this one neat trick of reformulating the acceleration with the chain rule.”

What a wonderful feeling. It was a piece of fun the two shared without any reservations towards the trivialities of the mundane. There lay a purpose in every action. Whether it was in his control or predetermined from the beginning of time, Theodore knew he was heading in the right direction.