Angular momentum L=rxP and L=I x omega ?

[LightQuanta]

When do we use L=r x P and L=I x Omega (angular velocity)?

in old 8.01x - Lect 24, I pasted here link of the lecture, which will take you at exact time (at 27:02)he says "spin angular momentum" in classical physics lecture and why? I expected to hear "angular momentum" vector.



Normally, "spin angular momentum" we deal with it in quantum mechanics.

So, how should I understand this correctly when to use L with moment of inertia or when to use L with r x P? I know both dimensionally equal.

My current understanding is that, I would use L with moment of inertia, when I see object spinning with mass isolated itself with different 3d solid or hollow geometry. Since we have each formula for respective moment of inertia.

I would use L= r x P when I see planetary motion in orbits objects separated by distance "r" or to prove Kepler's second law.

Can we independently take different vectors using right hand rule (individually) and combine actual direction of torque, angular momentum, etc into one diagram of cross products ? This combination in itself is a new vector perpendicular to plane of two vectors (taken from right hand rule) ? although they do not form a formula in combination ?

 

[wrobel]

Well, there are at least four theorems about angular momentum.

Let ##O\boldsymbol e_x\boldsymbol e_y\boldsymbol e_z## be an inertial frame and let ##A_1,\ldots,A_N## be a system of mass points with masses ##m_1,\ldots, m_N##.
Assume also that ##\boldsymbol F_k## is an external force that applied to the point ##A_k##. Then
$$\boldsymbol L_0=\sum_{i=1}^Nm_i\boldsymbol{OA}_i\times \boldsymbol v_i;\quad \boldsymbol\tau_0=\sum_{i=1}^N\boldsymbol{OA}_i\times\boldsymbol F_i$$
and $$ \frac{d}{dt}\boldsymbol L_O= \boldsymbol\tau_O\qquad (1).$$
By ##S## denote a center of mass of this system. Let ##S\boldsymbol e_x\boldsymbol e_y\boldsymbol e_z## be a moving coordinate frame
that have ##S## as the origin and does not rotate. By ##\boldsymbol v_i^r## denote a velocity of the point ##A_i## relative to ##S\boldsymbol e_x\boldsymbol e_y\boldsymbol e_z##.

Then
$$\boldsymbol L_O=m\boldsymbol{OS}\times \boldsymbol v_S+\boldsymbol L_*,\quad \boldsymbol L_*=\sum_{i=1}^Nm_i\boldsymbol{SA}_i\times \boldsymbol v_i^r,\quad m=\sum_{i=1}^Nm_i;$$
and
$$ \frac{d}{dt}\boldsymbol L_*= \boldsymbol\tau_*,\quad \boldsymbol\tau_*=\sum_{i=1}^N\boldsymbol{SA}_i\times\boldsymbol F_i.\qquad (2)$$

If the system of mass points ##A_1,\ldots,A_N## forms a rigid body with fixed point ##O## then formula (1) remains valid with ##\boldsymbol L_O=J_O\boldsymbol\omega,## where ##J_O## is the inertia operator about the point ##O##. ##\boldsymbol\omega## is rigid body's angular velocity. The following formula is also true
$$\frac{d}{dt}\boldsymbol L_O=J_O\frac{d}{dt}\boldsymbol{\omega}+\boldsymbol\omega\times J_O\boldsymbol\omega.$$
For general motion of the rigid body formula (2) is used with
$$\boldsymbol L_*=J_S\boldsymbol\omega,$$
and $$\frac{d}{dt}\boldsymbol L_*=J_S\frac{d}{dt}\boldsymbol{\omega}+\boldsymbol\omega\times J_S\boldsymbol\omega.$$