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.
Well, the time is going to change from just t to t plus epsilon, and as a result, the Lagrangian is also going to change. So the new Lagrangian will be L prime, which is equal to the old Lagrangian , plus how much the Lagrangian changes over time, that's just dL by dt, multiplied by how long that change lasts, so multiplied by epsilon.
And it can go over here and go and so on and so on. the correct Lagrangian so you get the right action and out come the laws of physics.
so this is gonna become mv squared. And then we can sub in the Lagrangian . So this becomes minus 1/2 mv squared minus v,
And it can go over here and go and so on and so on. So you've got a separate Lagrangian for classical mechanics, for special relativity, for electrodynamics, and so on.
Previously on Veritasium, we learned that everything always follows the path that minimizes a quantity known as the action. This is equivalent to the integral of the Lagrangian L over time. In the simplest case, that's just the kinetic minus potential energy.
Well, remember that in the simplest case, the Lagrangian is just equal to the kinetic minus potential energy, which we can write as 1/2 mv squared minus v. So if we take the partial derivative of the Lagrangian with respect to v, we're just gonna get d over dt, m times v multiplied by v, so this is gonna become mv squared.
If you want to, you can play around with this. What does the Higgs mass parameter in the standard model Lagrangian look like, knowing its mass?
It's, on the one hand, a fantastic achievement, because there hasn't been one single experiment which could not be described actually by this formula, or whatever, the Lagrangian , you call it, of the standard model. You can simplify it a little bit.
In fact, we made a whole video on this over a year ago, but for now, all we need to know is that 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.
So what is it? Well, remember that in the simplest case, the Lagrangian is just equal to the kinetic minus potential energy, which we can write as 1/2 mv squared minus v. So if we take the partial derivative of the Lagrangian with respect to v, we're just gonna get d over dt, m times v multiplied by v,
- So suppose we do an experiment where the result is the same now as some tiny time interval epsilon later, then how does this affect the action? Well, the time is going to change from just t to t plus epsilon, and as a result, the Lagrangian is also going to change. So the new Lagrangian will be L prime, which is equal to the old Lagrangian , plus how much the Lagrangian changes over time,
And it can go over here and go and so on and so on. The thing that will encompass all of physics in reality, what people are asking is what is this Lagrangian that can spit out all of the laws of physics in this universe?
So make you guys hungry. Now, Earth. Sun. Lagrangian place.