stumbled upon it years ago but I’m terrible at math so kinda forgot about it for years, recently stumbled upon it again so out of pure curiosity, what is it? does it have real world use or is it something purely theoretical?
In: Mathematics
I don’t have the knowledge base to explain the what, hit I can do the uses! I guess for ELI5, the Laplace transformation is where we take normal math and turn them partly imaginary so the math is easier. My professor said that, “The shortest path between two systems of equations is through the imaginary plane!”
Imagine your car going down the road, you want that car to be nice and steady, so we have suspension systems. This suspension might look like the mass of the body (how much it weighs), a spring (lets the car body bounce up and down), a damper (makes the body move a little less when it’s going up and down), and the street it’s driving on. We know what we want the movement of the car to look like, where the spring bounces us gently and the damper gets us back to normal so we don’t constantly bounce. So we want something to relate the force coming in to our Change in movement, x.
This is a tough problem to math out directly. The damper converts motion to heat, so we can’t add up our energy. We can test a system by applying a force to it, and then the car bounces with changing acceleration, velocity, and position! Currently we have an input and we get three related outputs (as velocity and acceleration are derivatives of the position), and we don’t have the tools to solve this with differential equations. That’s where Laplace transforms come into play. Newton’s second law (sum of forces = mass times acceleration) will have our input force (f), position (spring force kx), velocity (damp force cx’), and acceleration (mx’’), but the transforms for derivatives all have the same function big X (something like mXs^2 + cXs + kX = F). The s is a new, complex variable. It involves real and imaginary numbers, but we don’t have to worry about its value!
Next comes the hard math part, as we have to rearrange our equation so the pieces fit some Laplace transform. We can un-distribute X (F = (ms^2 + cs + k)X), isolate our output (X = 1/(ms^2 + cs + k) * F), and find some similar transform to use. We might use something like b/((s-a)^2 + b^2); we arrange 1/(ms^2 + cs + k) to fit whatever form we pick, and then we can apply the inverse transformations to get rid of the s variables and get back into real space. The result is that we transformed an equation built off acceleration, velocity, and position as well as our inputs, and we can strip our acceleration and velocity.
This doesn’t make math easy, but it makes math easier. When we don’t have enough equations to solve it with system algebra, we can use transforms as a shortcut to get it into the form we want. It can also be used for equations where we have different forms, like sinusoidal, exponential, and so on, and make them easier to solve with simpler tools.
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