Let’s solve the free particle

I guess

Newton’s Law

Hence

At least that works. Not sure I derived it particularly. Or proved it unique.

Whatever. Lagrangian version

Euler Lagrange Equations

How do you get that? By varying the action with fixed endpoints it’s the one that minimizes the path.

Nice.

Okay. What about the quantum version?

Well

How do I know that? In particular it’s hard to remember where the i goes. Well, I memorized it at some point. It follows that

But what is

Whatever. I’m bored.

Maybe I’ll do the path integral some other day