It’s true. I don’t normally write about politics and stuff like that. But you know what, these aren’t normal times. I have to write something — for me, it’s just basic therapy. So here we go.
Maybe you haven’t heard — but there’s a presidential election coming up. Yup. It’s true. It’s essentially a choice between Biden and Trump. I will let you make up your own mind on these two candidates — but I have to talk about Trump flags.
Yes, you see these flags. People put them on their trucks (but never on their hybrid-electric cars), boats, and houses. I’ve seen several different flags and I think I should give my translation for these things. …
It seems like the most common introduction to the Calculus of Variations is to talk about the Brachistochrone problem. It goes like this.
Imagine you have a bead that can slide along a frictionless wire under the influence of gravity only. You want the bead to slide (starting from rest) from point 1 to point 2 along some path. What path would produce the shortest time?
So, let’s do this.
Bead on a Straight Wire
I’m going to start with the simplest path possible — a straight line. It looks like this.
Oh, why did I pick weird starting and ending points? You will see why soon (hopefully). But here is how to find this sliding time. I’ll start by breaking the path into small pieces — where each piece has a length of ds. I need to find the time it takes the bead to move along this short piece. That time depends on the velocity. So, using the definition of the velocity, I can find the time. …
Here is a physics problem. You have an object above the moon and you drop it. What is the final speed before the object hits the surface? Note: this object is NOT the lunar lander since it also has a horizontal motion — but you get the idea. The best way to solve this problem is to use the gravitational potential energy (with respect to infinity) of the object-moon system. This potential looks like this.
In this expression, r is the distance from the center of the moon to the object and G is the universal gravitational constant (6.67 …