Question Statement (from Kleppner and Kolenkow "Introduction to Mechanics"):

Let us change variables as follows:

$sin u = sin(θ/2)/ sin(θ_0/2)$

The motivation for this is that although $ θ$ is periodic, as the pendulum swings through a cycle, θ varies between $−θ_0$ and $ θ_0$. On the other hand, $u$ varies between $−π$ and $+π$.

How come $u$ varies between $−π$ and $+π$ ? If we replace $θ$ with $−θ_0$ and $ θ_0$ in the given equation we get the value ranging from -1 to +1 i.e angle ranging from $−π/2$ and $+π/2$.

I was also confused why we could suppose "$sin u = sin(θ/2)/ sin(θ_0/2)$"? Elliptical integral is totally new to me.

Question Statement (from Kleppner and Kolenkow "Introduction to Mechanics"):

Let us change variables as follows:

$\sin u = \sin(θ/2)/ \sin(θ_0/2)$

The motivation for this is that although $ θ$ is periodic, as the pendulum swings through a cycle, θ varies between $−θ_0$ and $ θ_0$. On the other hand, $u$ varies between $−π$ and $+π$.

How come $u$ varies between $−π$ and $+π$ ? If we replace $θ$ with $−θ_0$ and $ θ_0$ in the given equation we get the value ranging from $-1$ to $+1$ i.e. angle ranging from $−π/2$ and $+π/2$.

I was also confused why we could suppose "$\sin u = \sin(θ/2)/ \sin(θ_0/2)$"? Elliptical integral is totally new to me.

Question Statement:

Let us change variables as follows:

$sin u = sin(θ/2)/ sin(θ_0/2)$

The motivation for this is that although $ θ$ is periodic, as the pendulum swings through a cycle, θ varies between $−θ_0$ and $ θ_0$. On the other hand, $u$ varies between $−π$ and $+π$.

How come $u$ varies between $−π$ and $+π$ ? If we replace $θ$ with $−θ_0$ and $ θ_0$ in the given equation we get the value ranging from -1 to +1 i.e angle ranging from $−π/2$ and $+π/2$.

I was also confused why we could suppose "$sin u = sin(θ/2)/ sin(θ_0/2)$"? Elliptical integral is totally new to me.

Question Statement (from Kleppner and Kolenkow "Introduction to Mechanics"):

Let us change variables as follows:

$sin u = sin(θ/2)/ sin(θ_0/2)$

The motivation for this is that although $ θ$ is periodic, as the pendulum swings through a cycle, θ varies between $−θ_0$ and $ θ_0$. On the other hand, $u$ varies between $−π$ and $+π$.

How come $u$ varies between $−π$ and $+π$ ? If we replace $θ$ with $−θ_0$ and $ θ_0$ in the given equation we get the value ranging from -1 to +1 i.e angle ranging from $−π/2$ and $+π/2$.

I was also confused why we could suppose "$sin u = sin(θ/2)/ sin(θ_0/2)$"? Elliptical integral is totally new to me.