Thread by @astroparticular

🎥 Ready, set, ACTION!

Here's a still from one of my favorite movies ever: Interstellar. But what is Murphy writing on the board? What does this equation mean?

Luckily, this gives me a reason to talk about one of my favorite concepts in physics: the principle of least action!

Instead of looking at Murphy's really complex (ten dimensional action), let's look at a simpler version.

This equation below is the definition of action.

Let's pick it apart together and understand why it's so powerful. 💪🏽

📖 The function L is called the "Lagrangian." It describes the energy in a system where L = T - U, the difference of the kinetic and potential energy.

The Lagrangian describes the trajectory of a particle: how it moves and what forces it responds to.

📖 Now S is the action. To get S, we have to sum over many different possible trajectories. It has a very special property that we'll see later!

🙌🏽 Putting it all together, the action sums over all of the possible paths a particle can take.

Why is this important? Well, what if we want to determine the shortest path a particle will take given its circumstances? 🤔

🤓 To find out the shortest path, we can pull out our calculus and minimize the action; that is, find when its derivative is equal to zero.

💪🏽 Math ahead; let's grind through.

✍️ We can describe how the action changes by writing the Lagrangian in terms of its parameters q and qdot, and how those components change.

Now we're gonna separate this into two integrals and presciently calculate the right-hand integral first.

Let's use integration by parts! ✍️

Now we're left with a boundary term in the middle.

In the case that we look at two fixed endpoints, we can set this boundary term to zero because dq should converge at the endpoints of our path.

But notice how both integrals have dq now. We can put them back together again! 🔥

Then when we put the integral back together again, we find an equation that lets us minimize the action. What's this equation?

🎉The Euler-Lagrange Equation! 🎉

💪🏽This is a super powerful equation. It lets us minimize the action so long as we know the Lagrangian of a system!

Let me show you how powerful it is by showing you that we can derive Newton's equations from it.

To use the Euler-Lagrange equation, let's feed it a Lagrangian.

Let's use this really standard one with a familiar kinetic energy term and a potential energy term.

If we plug in the Lagrangian above into the Euler-Lagrange equation, we get Newton's equation from it. F = ma.

It seems that Newtonian physics is encoded into the Principle of Least Action! 🤯

🙌🏽 Wow! The concept of action can tell us a lot about physics! (It even does so in GR and QFT) 🙌🏽

This is why when Murphy wrote this equation in Interstellar it was so important.

Curious how we can learn so much from minimizing a function(al)! 😲