I was recently learning about the calculus of variations and came across the euler lagrangian formula. $$L_y - \frac{d}{dx}L_{y'} = 0$$ Where $L$ is the Lagrangian
While learning how to use it from a video of Alexander Fufaev, he explained that using an analogy of the displacement-time graph of a ball thrown upwards
He told that the parabola which is formed by the ball is due to the property of the nature and that nature is extremal.
But according to me, i think that the reason the ball follows that exact parabola, is relalated to its position, velocity, and acceleration. This is relevant as $$L=L(x,y,y')$$ But also the primary reason is that it doesn't have any other force acting on it which doesn't cause it to have any other weird graph and follows the well defined parabola. And Fufaev tells that it's a property of nature that it will follow the lowest Lagrangian. Which means that if the Langragian of the path of a motion is high, Nature will Somehow, exert a force, to change its lagrangian to a lower state. But this seems totally counter intuitive as how can nature apply a force itself? Just to change the motion of a body?
I am a ninth grader and all of this is self learnt. So try to give me a more imaginative and visual solution rather than a bunch of math formulae.
Fufaev's video here
