Consider the point mass $M$ being hold by a massless string at distance $r$ and rotate at angular frequence $w$.
Energy:
To orter to obtain the energy of the point mass, we calculate as a rotational motion: $E_{rotational}=\frac{1}{2}Iw^2=\frac{1}{2}mr^2w^2=\frac{1}{2}mv^2=E_{kinetic}$.
Thus the rotational motion and kenatic energy is exactly equivelent(by dm).
Momentum:
$|L|=|r\times P|=|rmv|=|rP|$ so there was a difference between angular momentum and linear momentum, where from Mike Dunlavey's answer in (How can linear and angular momentum be different? ) and wiki (https://en.wikipedia.org/wiki/Angular_momentum#Discussion), I kind of get the picture that linear momentum is the dipict of conservition of the body's motion in general, where the angular momentum is the body's motion "at distance".
So when I look back at the energy part, it's saying that linear momentum with respect to the "center of the motion", where angular momentum was with respect to the mass at "length" or say "rotational arms".
However, I still have some questions:
I want to make sure that if there are some mistakes in the above statement.
Could you give me some idea about your understanding of the difference between angular momentum and linear momentum?
Is there any way to write linear momentum and angualr momentum by the same equation?(Unify them with a single integration/differential equaion)