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.
| Term | Meaning | Explanation |
|---|---|---|
| $\dot{I}_{ij}$ | Time Derivative of the Inertia Tensor | This represents how the inertia tensor changes over time due to the rotation of the coordinate system fixed in space, but not necessarily fixed to the body. For a rigid body, $I_{ij}$ is constant in the body's frame, but it changes in the stationary frame as the body rotates. |
| $I_{ij}$ | Moment of Inertia Tensor | This $3 \times 3$ tensor defines the mass distribution of the rigid body and its resistance to angular acceleration. |
| $\omega_j$ (and $\omega_{\ell}$) | Angular Velocity Vector | The components of the vector $\vec{\omega}$, which describes the angular speed and axis of rotation. |
| $\epsilon_{ijk}$ | Levi-Civita Symbol | The completely antisymmetric tensor of rank 3. It is key to expressing the cross product ($\vec{A} \times \vec{B}$) in index notation, which is the mathematical representation of rotation. |
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 ($\epsilon$).
What is the rotational identity for the moment of inertia tensor-L.mp4