#laplace transform

OKAY SO

once upon a time this guy came out of a lab experiment gone wrong-

Wait wrong Laplace Transformation!

So!

The Laplace transformation is a very helpful transformation that, from my brief experience with it in dif eq, basically takes in differential equations, and turns them into algebraic equations that are easier to solve.

The way the Laplace transformation does that, is it inputs a function f(t), and it outputs a function F(s) = L{f(t)}(s) that's easier to work with. The trade off is, while t is a real number, s is now allowed to be a complex number (so, wider domain, but we get a nicer function!). The other trade off is, once we've solved for whatever variable we're looking for in the complex domain (in terms of s), we now have to convert our variable back into what it is in the time domain (real numbers, in terms of t).

This fun doohicky defines how we get our output F(s) from our input f(t). This is the Laplace Transform:

Basically what this does, is it takes your f(t) function, multiplies it by this e^(-st) function (which gets exponentially smaller as t goes out to infinity for any fixed s), and then finds the area underneath your function from t = 0 to infinity, in terms of s.

That's a mouthful. And not very helpful.

What it does that makes life so much nicer in differential equations, is it takes all your functions and derivatives of functions and derivatives of derivatives of functions, and it turns them into algebraic equations! Equations that can be solved with normal algebra stuff, rather than trying to directly figure out what your original function is given some relation to do with its derivatives!

In the link I linked (Wikipedia is a good source for math things when you half-know what you're talking about), there's a table that goes into how Laplace Transforms transform your function, the derivative of your function, other manipulations of your function, etc. So! Let's play with an example!

Suppose we have a damped spring. A spring where one end is fixed to the ceiling, the other end is fixed to a weight, and we pull the weight down and watch it bounce until it slowly stops bouncing.

The differential equation describing this motion is:

where m, c, and k are just numbers... but f(t) is the function we're looking for, f'(t) is its derivative (slope of f(t)), and f''(t)) is the derivative of f'(t) (slope of f'(t), or acceleration of f''(t)).

We have ways of dealing with this equation.

Those ways effectively come from the Laplace transformation!

So, from this handy dandy table in another Wikipedia article, we have the following tools:

This gives us the following, when we Laplace Transform that damped spring:

"THIS IS UGLIER AND LESS SIMPLE!" I hear you cry.

But guys - guys, trust me on this - guys wait! There are no more functions that are derivatives of the original function!

That f'(0-)? That's a constant!

That f(0-)? Also a constant!

And yes, s is a variable, but it's just s, not an unknown function!

The only actual function we have to solve for is F(s), guys!

I know, I know it doesn't look great now, but trust me - trust me, it'll look better! We can actually solve for what F(s) is, where before we couldn't solve for what f(t) was!

So, what do we get for F(s)?

Well, it looks a little funky, but it actually works really nicely when you actually have m and c and k and your initial conditions (f(0) and f'(0)) as numbers:

Like I said! Looks funky! But we can say that m*f(0) is some number a, and (m*f'(0) + c*f(0)) is some number b, and we get:

And THAT can be rewritten as a version of the sum of two functions that looks like this:

I know, I know, I'm making it look complicated again - but there's a REASON being the MADNESS!! Look at the table! Look! Look!!

That first bit multiplied to A turns back into a function in the time domain! And so does the second bit, multiplied by B!

So, NOW we have finally solved for f(t), and we have, for t greater than 0:

Which is kind of what we expect! We've got cos and sin with period wt giving us our bouncy spring with their wave function stuff, and we've got e^(-at) giving us the fact that as time goes on, the bouncies decrease exponentially until there are no more bouncies, with the exponential decay!

The math matches the physics of the situation! Isn't that cool?

And because we could solve explicitly for this using the Laplace transform, we know that there aren't any hidden bells or whistles!

Precious few of you know what cursed and evil Discord conversation this was for, back in 2023. Despite the choices of letters (and case), this isn't in Laplace space, nor is it about electric current—it's just one real value that varies with some other real value squared.