# Inertia Tensor as a Transformation

• uliuli
In summary, the conversation discusses the relationship between angular velocity and angular momentum for a rigid body. It is mentioned that the inertia tensor can be rewritten using a rotation matrix and a diagonalized inertia tensor. The discussion then moves towards the transformations that can be performed on the angular velocity and the resulting effect on the angular momentum. It is noted that there is a lot of freedom in these transformations, with the only restriction being a positive inner product between the two vectors.
uliuli
Hi everyone,

I was thinking about the relationship between angular velocity and angular momentum for a rigid body: $I \omega = L$. In particular, I'm trying to gain a little bit of intuition as to what transformations $I$ can perform on $\omega$.

Let's use a reference frame at the center of mass of the body. We can rewrite the inertia tensor as: $I = R I_0 R^T$, where $I_0$ is the diagonalized inertia tensor and $R$ rotates the coordinate system from a principle-axes-aligned frame to the current frame. When I now apply $I$ to $\omega$, I am equivalently rotating $\omega$ by some arbitrary rotation, scaling each component by a positive (and in general, different for each component) number, and rotating by the inverse of the first rotation. In general, what does this concatenation of transformations give me?

Thanks!

To reply to my own thread :):

The more I think about this, the less I think one can say about the family of transforms that $I$ can perform. I can 'design' $I_0$ to be arbitrarily close to a projection onto a cartesian axis, I can make $R$ any rotation I want... there is a lot of freedom.

About the only concrete restriction on the transform I've come up with is that $\omega$ and $L$ have a positive inner product: $\omega^T L = \omega^T R I_0 R^T \omega = (R^T \omega)^T I_0 (R^T \omega) = \tilde{\omega}^T I_0 \tilde{\omega} > 0$ due to the positive definiteness of $I_0$.

Ohh well!

## What is the inertia tensor?

The inertia tensor is a mathematical representation of the distribution of mass within an object. It describes how the mass is distributed in three-dimensional space and how it will respond to external forces and torques.

## How is the inertia tensor calculated?

The inertia tensor is calculated by taking the mass of each individual point in an object and multiplying it by the square of its distance from a chosen axis. These values are then summed and organized into a 3x3 matrix.

## What is the significance of the inertia tensor in physics?

The inertia tensor is an important concept in physics as it allows us to understand how objects will behave when subjected to external forces and torques. It is used in various fields such as mechanics, engineering, and robotics to analyze the motion and stability of objects.

## How does the inertia tensor relate to rotational motion?

The inertia tensor is directly related to rotational motion as it helps to determine an object's moment of inertia, which is a measure of its resistance to rotational motion. The inertia tensor can also be used to calculate the angular momentum of an object.

## Can the inertia tensor be used for any type of object?

Yes, the inertia tensor can be used for any object, regardless of its shape or size. However, for irregularly shaped objects, the calculation may be more complex and require advanced mathematical techniques.

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