The velocity vector $\vec{v}$ (or its component $\dot{x}_i$) is related to the angular velocity $\vec{\omega}$ and the position vector $\vec{x}$ by the fundamental equation for a point in a rigid body rotating about a fixed point: $\vec{v} = \vec{\omega} \times \vec{x}$

In the context of the proof you're viewing, this cross-product is expressed in component form using the Levi-Civita symbol ($\epsilon_{ijk}$) and the Einstein summation convention (where repeated indices are summed over).

Component Relationship

The component form for the $i$-th component of the velocity $\vec{v}$, which is $\dot{x}_i$, is:

$$ \dot{x}_i = (\vec{\omega} \times \vec{x})i = \epsilon{ipq}\omega_p x_q $$

• $\dot{x}_i$ is the $i$-th component of the velocity vector $\vec{v}$.
• $\omega_p$ is the $p$-th component of the angular velocity vector $\vec{\omega}$.
• $x_q$ is the $q$-th component of the position vector $\vec{x}$.
• $\epsilon_{ipq}$ is the Levi-Civita symbol, which is 1 for an even permutation of ($i, p, q$), -1 for an odd permutation, and 0 if any two indices are the same.
• The repeated indices $p$ and $q$ imply a summation over all $p, q \in \{1, 2, 3\}$.

This expression allows the proof to substitute the velocity $\dot{x}k$ in Step 2 with terms involving $\omega_p$ and the position coordinates $x_q$, which is essential for relating the time derivative of the inertia tensor ($\dot{I}{ij}$) to the term $\epsilon_{ijk}\omega_j I_{kl}\omega_l$.

Brief audio

How is the velocity vector related to the angular velocity and position in the proof-L.mp4