In the simplest case, that's just the kinetic minus potential energy. Euler and Lagrange found that the principle of least action is obeyed, so long as this set of differential equations is satisfied.So Noether used action to see how physics was affected by different symmetries.
Einstein used thought experiments to come up with his theory of relativity. Euler 's solution to the bridges of Königsberg puzzle is what inspired graph theory.And when Schrödinger wanted to illustrate his problems with quantum mechanics, he imagined a cat in a torture box.
And that's good enough. Euler seems to have been having fun.
And that's good enough. Euler said, I can prove a formula that relates the even values of the zeta function to the Bernoulli numbers.
Can you guess who? - Leonard Euler ? - Leonard Euler . - The two quickly became friends, bonding over a shared obsession with number theory.
Now Leonhard Euler proved an important formula, an important theorem, based on these circle functions relating complex exponentiation to cosine and sine. Now Euler was one of the great mathematicians of all time. And so you can bet that if something is called Euler 's formula, it must be pretty important, because he proved a lot of formulas.
Just by the nature of space and time, everything gets stretched and squeezed into an infinitely thin line. It's called Euler 's disk for no good reason.
Just by the nature of space and time, everything gets stretched and squeezed into an infinitely thin line. And this Euler 's disk takes a little while to settle down, but in its final stages,
And if you haven't, well, you're about to get an introduction. So to talk about Euler 's identity, I want to discuss the circle functions. So let's look at what's called a unit circle, radius equals one.
So that point has x-coordinate cosine theta, and y-coordinates sine theta. Now Leonhard Euler proved an important formula, an important theorem, based on these circle functions relating complex exponentiation to cosine and sine. Now Euler was one of the great mathematicians of all time.
Which brings me to what I think is not the most beautiful equation. Which is Euler 's formula evaluated when theta equals pi. So again, looking at our circle, when theta equals halfway around the circle, cosine pi is negative 1, sine pi is zero,
has references to p versus np, that great unsolved problem. It has references to Euler 's equation again, has references to the Utah teapot. It has stuff in ASCII.
This portrait that's hanging there on the wall not too far from the hospital cafeteria was assembled by kids who were in the hospital in summer This is an Eulerian, a connected Eulerian subgraph that looks like the letter E.
This portrait that's hanging there on the wall not too far from the hospital cafeteria was assembled by kids who were in the hospital in summer So I have an Eulerian circuit that looks like the Konigsberg Bridge Problem.
However, they'll do it in three dimensions. And you can see our Euler spirals now where they evolve from the plane upwards. And this curve, we defined it to be the curve that penalizes for the change, both in curvature and in torsion.
- If this does not blow your mind, then you have no emotion. I thought you all would appreciate Euler 's identity. But as I mentioned, it's much more than just videos.
is the same as multiplying by negative 1. So the original form of Euler 's identity has a transparent geometric meaning that's obscured when we write it in terms of pi. So I believe this really ought to be called Euler 's identify.
So the original form of Euler 's identity has a transparent geometric meaning that's obscured when we write it in terms of pi. And we can relate this to Euler 's identity by noticing that e to the i tau equals 1 says be one with the tau.
This portrait that's hanging there on the wall not too far from the hospital cafeteria was assembled by kids who were in the hospital in summer And then the concluding pun is an Eulerian pun.
we can write down the kinetic minus potential energy to find what's known as the Lagrangian. Then you sub that in to the so-called Euler -Lagrange Equation, and out comes your solution. For example, predicting the motion of a double pendulum by using this standard forces approach is infamously hard.
And so that pendulum is in this moving reference frame as it's swinging. - But if you pluck the kinetic and potential energy into the Euler -Lagrange Equation, then you can quickly get to a solution at least numerically. That's actually how we made this simulation.
and he thought, "Well, with even numbers, perhaps just two prime numbers would be enough to sum to give you every even number." In fact, as Euler wrote in his letter, this is how Goldbach originally framed the problem in person. So, Euler reformulated Goldbach's idea into two separate conjectures.
In fact, as Euler wrote in his letter, this is how Goldbach originally framed the problem in person. So, Euler reformulated Goldbach's idea into two separate conjectures. The first dealt with odd numbers, and said, "Every odd number greater than 5 can be written as the sum of three primes."
If you can prove the weak conjecture, you still don't get the strong one. Now, after Euler reformulated these conjectures, he was so confident they were true, that he wrote, "I regard this as a completely certain theorem, although I cannot prove it." - And he couldn't, you know?
For 2D, if we wanted to work in 2D, we could go back to the very best, to curves that are called Euler spirals, who were discovered independently by Bernoulli, by Euler , and by Talbot to use them to design railway trucks. OK? And the nice property of this curve is that the curvature evolves linearly along the curve.
And another quote from Thomas Pynchon-- "Who claims truth, truth abandons." And there's a little historical context. And I think that-- was it Euclid or Euler who thought that there didn't even exist Graeco Latin squares beyond a certain single digit
Now Euler was one of the great mathematicians of all time. And so you can bet that if something is called Euler 's formula, it must be pretty important, because he proved a lot of formulas. So here it is.
So the original form of Euler 's identity has a transparent geometric meaning that's obscured when we write it in terms of pi. So I believe this really ought to be called Euler 's identify.
And that's good enough. And so on. So Euler showed that zeta of two is pi squared over 6.
This portrait that's hanging there on the wall not too far from the hospital cafeteria was assembled by kids who were in the hospital in summer Hamiltonian, Eulerian, Tree. I don't know.
What's really going on here? Well, we can find out by rewriting the Euler 's identity traditional form in terms of tau. It's e to the i tau over 2 equals negative 1.
It was not the exact form of the conjecture which now bears his name, but something related. "It seems that every integer greater than 2 can be written as the sum of three primes." Euler was intrigued by the proposition, but he thought the idea could be refined further.
times dv over dt. But we can sub in the partial derivative of L with respect to x with this term from the Euler -Lagrange equation. And we can simplify this further by writing dx over dt as v, and that gives us this expression.
- So some of the famous transcendental numbers would include the number pi, you know, the 3.14159265 and so on. So that's a transcendental number. Also, Euler 's constant, the e, like e to the x, the exponential function. - So you could say that some of the sexiest numbers in mathematics are all transcendental numbers?
And therefore, in in 2003 used them to complete images. We wanted to define a new type of curves, which we call 3D Euler spirals that will be very similar. However, they'll do it in three dimensions.
So this relates complex exponentiation to the circle functions. And in order to get what I believe deserves the name and what captures the true spirit of Euler 's identity, we're going to evaluate this formula when theta is equal to the circle constant, which is tau.
is like a huge growth area. You look an Ebay there's all these things that are worth hundreds of pounds is I sought this item here which looks like a hand grenade or a pepper grinder. with numbers and one of them the most famous one was Zacharias Dase who was employed by Euler actually it was either Euler or Gauss, maybe Gauss, where he calculated Pi to 200
This portrait that's hanging there on the wall not too far from the hospital cafeteria was assembled by kids who were in the hospital in summer So we've got H for Hamiltonian, E for Eulerian, T for tree.
You can do it if you're good, and people who are good at mechanics can do it. But with the Lagrangian approach, you could just write down the energy, which is a scalar not a vector, plug it into the Euler -Lagrange Equation, and you get the right equation to motion and you don't have to be a good physicist.
And for this problem, we're not working on images anymore, but we are going to do it in 3D on scanned object. For 2D, if we wanted to work in 2D, we could go back to the very best, to curves that are called Euler spirals, who were discovered independently by Bernoulli, by Euler , and by Talbot to use them to design railway trucks.
At this point, more mathematically sophisticated people are scrambling through their heads, like what are some other things. Like, oh, yeah, oh, yeah, what about Euler 's identity. So if you've heard of this before, you might already have thought of it.