In the case of the Standard Model Lagrangian, each of these symbols mean a lot, the equation itself is uses cute short hand tricks, and it doesn’t even describe dynamics on its own without tying in the Principle of Least Action. “2+2=4” means nothing to an alien until after you tell them what each of those symbols mean and how they’re being used. All equations must have some context before they do anything or even mean anything. All the relevant dynamics are there, but there’s a lot of unpacking to do before that becomes remotely obvious. In some sense this equation is compressed data. The equation of everything (except gravity). Notably, it doesn’t cover gravity, but be cool. So what’s the most (but not needlessly) complicated equation in the universe? Arguably, it’s the Standard Model Lagrangian, which covers the dynamics of every kind of particle and all of their interactions. And if you want to describe the dynamics of a system, Lagrangians are an extremely compact way to do it. Generally, when you look at the same dynamics applied over and over, the equations involved don’t get much more complicated (although their solutions definitely do). There are also Lagrangians for everything from orbiting planets to electromagnetic fields. From this single formula, you get the conservation of energy, conservation of momentum (when moving sideways), as well as the acceleration due to gravity. Rather than talking about kinetic energy and momentum and falling, you can just say “Dudes and dudettes, if I may, the Lagrangian for an object flying through the air near the surface of the Earth is, where m is mass, v is velocity, and z is height”. With this principle, a single Lagrangian can be used to derive many physical laws at once, so it’s a good candidate for equations that aren’t needlessly complex.įor example, you can sum up Newton’s physics almost instantly. The principle of least action says that the path a system will actually take has the least action. The Lagrangian gives every point on this picture a value and the total along an entire path is the “action”. Evidently, if you want an equation that genuinely needs to be complicated, you don’t need a complicated situation, you need complicated dynamics. So while the situation itself is complicated, the equation describing it isn’t. This is just Newton’s law of gravitation,, repeated for every possible pair of stars and added up. In practice this is absurd (there are a few hundred billion stars in the Milky Way, but we can’t see most of them because there’s a galaxy in the way), but the equation you would use is pretty straight forward. If you want to do something like, say, describe the gravitational interactions of every star in the galaxy, you’d do it by numbering the stars (take your time: star 1, star 2, …, star n), determine the position and mass of each, and, and then find the force on each star produced by all the others. But like π, it is deceptively so (we hope). If you want to describe the behavior of a ball flying through the air it’s not enough to say “it goes up then down” there’s a minimum amount of math that goes into accurately calculating the path of falling objects, and it’s more complicated than that.Īrguably, the universe is pretty complicated.
If you’re looking for an equation that needs to be complicated, a good place to look is physics (I mean, what else do you really need math for?). This is the basic idea behind “ Kolmogorov Complexity“ the length of the shortest possible set of written instructions that can produce a given result (never mind how long it takes to actually compute it). You can describe it simply and in such a way that anyone (with sufficient time and chalk) can find as many digits of π as they like.
Like π itself, this sum goes on forever, but it isn’t complicated. For example, This equation goes on forever, but it’s fairly straight forward: every term you flip the sign and increase the denominator from one odd number to the next. There are plenty of equations that are infinitely long, but often they’re simple enough that we can write them compactly. A better question might be “what is the most (but not needlessly) complicated equation?”. So if you really want to see the most complicated equation ever, call in sick for a few weeks. Physicist: If you have plenty of chalkboard space and absolutely nothing better to do, you can write down numbers, letters (Greek if you’re δυσάρεστος), and mathematical operators and eventually you’ll have the longest equation ever written down.
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