I just happened across this on YouTube today. I don’t recall ever hearing about this “paradox” before, but apparently it has sparked some heated debate over the years.
The Dome Paradox: A Loophole in Newton’s Laws
John Norton’s 2003 paper: (Microsoft Word - 003004.doc)
Here is the main part of his argument: The Dome
This is one of David’s recent papers that he shared with me. The finitist approach to physics has increasing adherants.
It’s remarkable that this individual is professor at a reputable university. Aside from the many unimportant errors in this paper, there is a fundamental misunderstanding of calculus. Prof. Norton integrates the equation of motion for this particle without accounting for a constant of integration: velocity is the integral of acceleration with an undetermined constant. That constant is also known as an initial condition in physics. He specifies one initial condition, r(0)=0 but implicitly makes the initial velocity nonzero, viz. r(t) = (1/144) (t-T)^4, which has a discontinuous derivative at t=T. Well, surprise, surprise: if the initial velocity is not zero, the particle will move!
tl;dr
Newton’s equation of motion requires initial conditions of position and velocity because it is a second-order differential equation.
My real motive for posting this comment is that it gives me the opportunity to tell my favorite calculus joke:
Two mathematics professors are having a pint or two at the pub.
Prof. A: Most people don’t know anything about calculus. I bet our waitress doesn’t what the derivative of e^x is.
Prof. B: I wouldn’t be so sure about that.
A: We’ll see.
[Prof. A goes off to the men’s room.]
B: [to waitress] I want to play a prank on my colleague. When he gets back, he’s going to ask you some questions. Just answer “e to the x.”
[Prof. B returns]
A: [to waitress] What is the derivative of e^x?
Waitress: e^x
A: What’s the integral of e^x?
Waitress: e^x… plus an arbitrary constant.
The waitress in the joke has a better command of calculus than Prof. Norton.
