The journey through these four simulations illustrates how index notation transforms abstract mathematical identities into a powerful tool for physical insight. By progressing from the fundamental Levi-Civita symbol in the basic cross product to the Kronecker delta in the complex moment of inertia tensor, we see how indices manage spatial relationships and mass distribution. The transition from a tilted, "wobbling" asymmetrical system to a perfectly aligned diagonal tensor demonstrates that index notation doesn't just calculate values—it reveals the internal geometry of a rigid body. Ultimately, the alignment of the angular velocity and momentum vectors in the final animation proves that the "principal axes" of an object are the specific directions where its complex, multi-dimensional resistance to rotation simplifies into a direct, predictable response.
The Inertia Ellipsoid animation synthesizes the relationship between mass distribution and rotational dynamics into a single geometric volume, visually manifesting the tensor equation $x^T I_{i j} x=1$. By mapping the moment of inertia as a distance $1 / \sqrt{I}$ from the origin, the cyan surface reveals the "Principal Axes" as the ellipsoid's lines of symmetry, where the narrowest regions identify directions of maximum rotational resistance. The inclusion of a mass offset demonstrates the Parallel Axis Theorem, showing how an off-center rotation center deforms the ellipsoid and shifts its orientation. Ultimately, the constant misalignment between the angular velocity ( $\omega$ ) and the angular momentum ( $L$ )-which only resolves when $\omega$ aligns with the ellipsoid's axes-provides a clear physical proof that an object's momentum is naturally "pulled" toward its directions of greatest mass concentration.
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