# The relation between Huygens-Steiner and parallel axis theorems

1. Aug 23, 2011

### akoohpaee

Hi,

Let $I^{\mu}_{\xi}$ denotes the inertia tensor of a system of N particles; each with mass $m_{i}$ and $\vec{r_{i}}$ represents the corresponding position vector with respect to an arbitrary point P. $I^{\mu}_{\xi}$ can be written as:

$$I^{\mu}_{\xi} = \sum_{i=1}^N m_{i} ( r_{i}^{2} \delta_{\xi}^{\mu} - r_{i,\xi} r_{i}^{\mu})$$

Moreover, $\tilde{I}$$^{\mu}_{\xi}$ is the inertia tensor in a coordinate system in which center of mass is located at the origin, that is:

$$\tilde{I}^{\mu}_{\xi} = \sum_{i=1}^N m_{i} ( \tilde{r}_{i}^{2} \delta_{\xi}^{\mu} - \tilde{r}_{i,\xi} \tilde{r}_{i}^{\mu})$$

where $\vec{\tilde{r}_{i}}$ represent position vector with respect to center of mass. According to Huygens-Steiner theorem, there exist a relation of this form between $I^{\mu}_{\xi}$ and $\tilde{I}$$^{\mu}_{\xi}$:

$$I^{\mu}_{\xi} = \tilde{I}^{\mu}_{\xi} + \sum_{i=1}^N m_{i} ( R^{2} \delta_{\xi}^{\mu} - R_{\xi} R^{\mu})$$

where $\vec{R}$ is the position vector of center of mass in the original coordinate system (with point P at the origin). I tried to prove the parallel axis theorem on the basis of Huygens-Steiner theorem. I multiplied both sides of the latter equation by $n_{\mu}n^{\xi}$, where $\vec{n}$ is a unit vector that represents the direction of rotation, i.e., if $\vec{\omega}$ is the angular velocity vector, $\vec{\omega} = \omega\ \vec{n}$.

$$n_{\mu}n^{\xi} I^{\mu}_{\xi} = n_{\mu}n^{\xi} \tilde{I}^{\mu}_{\xi} + \sum_{i=1}^N m_{i} ( n_{\mu}n^{\xi} R^{2} \delta_{\xi}^{\mu} - n_{\mu}n^{\xi} R_{\xi} R^{\mu})$$

which gives:

$$I = \tilde{I} + \sum_{i=1}^N m_{i} ( n_{\mu}n^{\xi} R^{2} \delta_{\xi}^{\mu} - n_{\mu}n^{\xi} R_{\xi} R^{\mu}) \ (1)$$

$I$ and $\tilde{I}$ are moment of inertia in coordinate systems with origin fixed on P and center of mass, respectively, given $\vec{n}$ describing direction of rotation. Also, as $\vec{n}$ is defined as a unit vector

$$n_{\mu}n^{\xi} R^{2} \delta_{\xi}^{\mu} = R^{2}$$

We can write equation (1) as:

$$I = \tilde{I} + \sum_{i=1}^N m_{i} ( R^{2} - (n_{\mu} R^{\mu})^{2})$$

or

$$I = \tilde{I} + \sum_{i=1}^N m_{i} ( R^{2} - (\vec{n} \cdot \vec{R})^{2})$$

It seems that, it is a generalization of the parallel axis theorem to three dimensions: if our particles are distributed in a plane (hence a two dimensional distribution), and the direction of rotation is perpendicular to the resulting plane, ($\vec{n} \cdot \vec{R} = 0$) we get the following equation:

$$I = \tilde{I} + \sum_{i=1}^N m_{i} R^{2}$$

which is the parallel axis theorem.

- Is parallel axis theorem a notion which only mention to this two dimensional case? If yes, then what this three dimensional generalization is named (if it has been named at all)?

- It seems that, the parallel axis theorem concerns relations between the moment of inertia in an arbitrary coordinate system and the center of mass frame, given a direction of rotation. On the other hand, Huygens-Steiner theorem concerns relation between inertia tensors in these two frames. Is it a correct picture?

Here, I want to thanks for your attention and apologize if it was too long.

Best regards and thanks again,
Ali