I fail to see how the integral over dE should be equal to pi. I've tried multiple subsitution but I can't seem to find it.
Anyone out there that can shed some light on this?
Mechanics (Volume 1) by Landau en Lifshitz, Third edition.
The chapter, or paragraph, is called "Determination of the potential energy from the period of oscillation"
What am I doing wrong?
I don't know about the minus sign you get though senpai
First of, Thanks for helping!
Doesn't the minus sign come from the change in variables?
And what inverse function comes out of the integral in your case? Can't seem to read it properly.
Whenever you're faced with an integral in physics which is in terms of physical quantities, you're first step should almost always be some (linear) change of variables to a dimensionless parameter. Then the resulting integral which just equal some number multiplied by the physical constants, and whether you can solve the integral or not is usually of little importance to the physics. In this case, you have an integral over energy which depends on the two dimensionful energy values alpha (which I'll call a because I'm too lazy to latex) and U.
Now the easiest dimensionless change of variables is a linear one, z, such that z(E=U)=0 and z(E=a)=1, which you can figure out is just given by z = (E-U)/(a-U). Then you end up with a an integral with no parameters which you can look up or solve with a sin(t^2) subsitution.