Quantization and Cohomology (Week 2)
Posted by John Baez
Here are the notes for this week’s class on Quantization and Cohomology:

Week 2 (Oct. 10)  The Lagrangian approach to classical mechanics. Action as the integral of a 1form (prelude).
 Homework: A spring in imaginary time.
Last week’s notes are here; next week’s notes are here.
A spring in imaginary time, you ask? Read on…
In last week’s lecture we learned  rather abstractly  how the dynamics of particles was analogous to the statics of strings. This time we introduce the basics of the Lagrangian formalism: enough to make the analogy more concrete, by considering an example.
What’s a simple problem involving the dynamics of a particle? How about the motion of a thrown rock in a constant gravitational field? We all know it traces out a parabola.
What’s the analogous problem involving the statics of a string? It’s just the problem of determining the equilibrium state of a string hung between fixed endpoints in a constant gravitational field. Instead of calling it a “string”, let’s call it a “spring”. Imagine a spring stretched out with its ends nailed to two posts… what curve does it trace out?
As you’ll see when you do this homework problem, the analogy is very cute. But there’s a funny wrinkle  obvious if you think about it. The thrown rock arcs up and then back down. The hung spring curves down and then back up! There’s a minus sign somewhere…
And, we can understand this minus sign by treating the spring as a thrown rock in imaginary time. The sign comes from
$i^2 = 1.$
The idea of relating dynamics and statics using imaginary time is well known in quantum field theory  it’s called “Wick rotation”. But, it works in classical field theory too, and here we see it in an even simpler context: classical mechanics!
Re: Quantization and Cohomology (Week 2)
I assume we’re neglecting the weight of the string itself, because otherwise “we all know” the hanging string actually traces out a catenary. Well, that’s true by definition. Specifically it’s not the graph if a quadratic function, but of a hyperbolic cosine.
So if we neglect the weight of the string in the statics problem, what are we neglecting in the dynamics problem? Not the weight, or the rock would move in a straight line. Not the air resistance, because that wouldn’t make the path of the rock a catenary, and we’ve neglected air for the string so that would mean something else we’re neglecting for the rock.