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
In not-entirely-eli5 terms, it’s a mathematical operation that converts a function that varies over time to another function that varies over… something else. That something else is called the “laplace domain” and it is beyond my ability to ELI5 what it actually is. Well tbh the entire thing is not very ELI5-able.
Anyway, Laplace transform is an extremely useful tool because it turns complicated and difficult integrations and differentiations into much, *much* simpler multiplications and divisions. That means doing math in the laplace domain is much easier. If you have one of the above-mentioned functions that vary over time, and you want to use that to do some complicated math, you can make things quite a bit simpler and faster by putting the function through a laplace transformation first. Afterwards you can compute the inverse of the transformation and get your result back as a function that varies over time. You get to the same end result, but you take a much faster route to get there. This all makes the laplace transformation a very commonly used tool in many engineering fields. Even computers can benefit quite a lot from the simplification that it provides.
Funny enough, just last night I ended up watching an anime that went into this a bit. Rampo Kitan: Game of LaPlace (less-than-stellar murder mystery series). So you can add inspiring a tv show to real-world uses.
I am not a math/physics person, so I cannot confirm/deny the actual accuracy of how anything is represented, but the plot involves using the transformations to create a LaPlace’s demon. The examples in-story may be a bit unrealistic, but might help with conceptualizing overall?
Do you know how if you wanted to add a large number of the same number, you could multiply once? And how instead of multiplying a number multiple times, you could find the exponent?
Differentiation is an operation we do to break up a continuous function into an infinite number of tiny tiny pieces.
Integration is the reverse operation and it builds the function back from the differentiated pieces. Now, this operation is easy for simple functions. But it can get hairy for difficult ones. And physics is full of very difficult integrals.
Laplace transform converts any function from real space into a complex space. In this complex space, differentiation and integration becomes multiplication and division.
So all of a sudden, you can complete your operations without having to figure out and solve complex differentials by transforming to the Fourier domain, complete the operations and transform back.
So the topic is a bit iffy for eli5 but can eli16
The basis of this is that you understand what an algebraic expression is. Like 4x = 20 you know x must be 5.
In calculus we care a lot about the slopes of functions. You’re probably familiar with something like y = 3x + 2 and it has a slope of 3 everywhere. A big part of calculus is finding the slopes of any functions at any point. The line example y = mx + b is taught in algebra because the slope is the same everywhere. But what about the slope of x^4 + 3x + 2?? Calculus can answer this.
Turns out slopes are useful for physics/science. If you’ve ever taken physics you spend a lot of time doing problems with newtons second law F = ma and they have you solving for force or acceleration all the time. People in the real world don’t care about acceleration all that much but what we do care about is the positions of things. We know that acceleration is the change in velocity over time. Or in other words, the slope of the velocity function. What’s velocity? It’s change in displacement over time (position with a direction). So most of the time if we want to know where a particle is located we can use high school physics to set up acceleration = something. Then we have to find a function where the slope of the slope is equal to that something. If that sounds hard, most of the time it is. But here’s where the Laplace transform comes in. If you take the Laplace transform or that equation, instead of dealing with slopes and calculus it turns the equation into an algebraic expression where you can use your seventh grade algebra skills to solve for a variable. Then you can do the inverse transformation to get back what you were looking for. The Laplace transformation and inverse transformations are things you rarely “do” but look up in a table due to patterns.
The tl;dr answer Is that the Laplace transform turns a differential equation into an algebraic expression.
The eli5 that glosses over a lot is it turns an equation you need to be a sophomore in college to solve into one you need to be a seventh grader to solve. It doesn’t always work nicely but it’s a cool trick to try when faced with a complicated problem.
If you’re interested in learning more, search “what is a differential equation” because it’s mainly a technique used to solve those.
Transforms generally take a vector from one vector space to another. The neat thing is that you can pack concepts like functions into your vectors and calculus operations like integration and differentiation into your concept of a vector space.
The LaPlace transform lets you take a function from the normal calculus universe and map it into one where integration and differentiation are represented by a variable that gets multiplied or divided in. This lets you complete what might be a difficult or out-right impossible integration with algebra.
Mathematical tool that allows you to convert differential equations into algebraic equations.
On other words, it make easier to solve differential equations
The Laplace Transform is “just” a disguised Fourier transform. They both do the same sort of thing: treat the space of functions as an infinite-dimensional vector space, and change to a different set of basis vectors. The new basis vectors happen to diagonalize the matrices that correspond to differentiation, I.e. they convert differentiation to multiplication (and vice versa). The Fourier transform uses oscillating functions (imaginary exponentials) as a basis for all functions, and the Laplace transform instead uses real exponentials. The two are exactly equivalent if you allow the frequency (of the Fourier transform) or the *s* parameter (of the Laplace transform) to be complex.
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.
The ELI5 answer is that it turns differential equations problems into simple algebra problems. Each derivative becomes a variable, so you can solve for the answer using relatively simple math, then convert it back to the original form and have a solved differential equation. It has lots of real world uses in engineering. Electric circuits, spring mass damper systems, and heat transfer equations can all be solved with LaPlace.
It transforms calculus into algebra. It is incredibly useful for math involving rotations and periodic functions – practically anything that has to do with radio, waves, electricity, etc. The math is much much much simpler.
In Electrical Engineering, we solved a simple circuit that had inductors and capacitors the long way – calculus every step of the way. It covered two whiteboards and took most of the class period. We then solved the same problem with the Laplace transform in the frequency domain. It took a couple minutes and most of that was writing the problem down – it turned the math into “add these numbers here, divide by this number here, …… and we’re done”
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