❓What is the rotational identity for the moment of inertia tensor?

The rotational identity for the moment of inertia tensor, which the document sets out to prove, is: $\dot{I}{ij}\omega_j = \epsilon{ijk}\omega_j I_{k\ell}\omega_{\ell}$

Here is an explanation of what the rotational identity represents and the meaning of its terms:

$\dot{I}{ij}\omega_j = \epsilon{ijk}\omega_j I_{k\ell}\omega_{\ell}$

This identity mathematically confirms that for a rigid body, the time rate of change of the moment of inertia tensor, $\dot{I}_{ij}$, is consistent with the body's rotation.

Physical Meaning

The identity relates the rate of change of the moment of inertia to the rotation itself. In a slightly simplified form, it can be viewed as an identity derived from the fact that the time derivative of any vector $\vec{A}$ fixed in a rotating body, as seen from a non-rotating frame, is given by:

$\frac{d\vec{A}}{dt} = \vec{\omega} \times \vec{A}$

The moment of inertia tensor, $I$, is a second-rank tensor. This identity extends the concept of the time derivative for a rotating vector to a rotating tensor, showing that the rotational effects are entirely captured by contracting the tensor with the angular velocity ($\omega$) and the Levi-Civita symbol ($\epsilon$).

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What is the rotational identity for the moment of inertia tensor? | FAQsviadean on Notion