Inertia tensor

The Inertia Tensor can be described as a general description of how the mass of a rigid body is distributed relative to all of the axes of distribution.

Inertia tensor takes the form of a 3x3 matrix.
It will change based on how you describe your coordinates.
Can be also thought of how “hard” it is to spin an object from a specific axes. Think of spinning a meter-stick at the end or the middle.

{I} = I_{11} I_{21} I_{31} I_{12} I_{22} I_{32} I_{13} I_{23} I_{33}

Where

I_{11} = α \sum m_{α} (k \sum x_{α, k}^{2} - x_{α, 1} x_{α, 1})

= α \sum m_{α} (r_{α}^{2} - x_{α, 1}^{2}) = α \sum m_{α} (x_{α, 2}^{2} + x_{α, 3}^{2})

{I} = \sum_{α} m_{α} (x_{α, 2}^{2} + x_{α, 3}^{2}), - \sum_{α} m_{α} x_{α, 2} x_{α, 1}, - \sum_{α} m_{α} x_{α, 3} x_{α, 1}, - \sum_{α} m_{α} (x_{α, 1} x_{α, 2}), \sum_{α} m_{α} (x_{α, 1}^{2} + x_{α, 3}^{2}), - \sum_{α} m_{α} x_{α, 3} x_{α, 2}, - \sum_{α} m_{α} x_{α, 1} x_{α, 3} - \sum_{α} m_{α} x_{α, 2} x_{α, 3} \sum_{α} m_{α} (x_{α, 1}^{2} + x_{α, 2}^{2})

This is the general inertia tensor for particles

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Inertia tensor

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