# Can the motion of a rigid body be decomposed into translational and rotational motion about ANY point?

My textbook mentions Chasles' theorem which asserts that

" it is always possible to represent an arbitrary displacement of a rigid body by a translation of its center of mass plus a rotation around its center of mass. "

Is the center of mass the only such point in general or can that be done for any point? I am aware that in some cases such as a rigid body pivoted at one end, the motion can be represented as (zero) translation of the pivot-point and rotation about that pivot, but is this true for any point in general?

• See here en.wikipedia.org/wiki/Screw_axis
– Eli
Commented Jul 10 at 7:13
• Any geometric point at rest with the body (even outside the body) has the requested property. Commented Jul 10 at 8:37
• @ValterMoretti, what does it mean, mathematically, for a point outside a moving body at to be at rest? Commented Jul 10 at 11:02
• Consider a triple of ortogonal axes $e_1,e_2,e_3$ at rest with the body centered at a point $P$ of the body. Any vector position that is a constant linear combination of these axes $v= \sum_{k=1}^3 x_ke_k$ detemines a point $Q=P+v$ in the rest space of the body. It does not matter if $Q$ is a material point of the body. Think of the center of a massive circumference... Commented Jul 10 at 11:53
• According to Michel Chasles' theorem, any arbitrary motion of a rigid body can be represented by a screw motion. In this motion, the body rotates about the instantaneous axis of rotation and simultaneously translates in the direction of the rotation axis. do you need the equations?
– Eli
Commented Jul 10 at 13:00

A rigid body $$B$$, by definition, is such that, for every point $$O\in B$$ there is a triple of orthonormal axes $${\bf e}_1, {\bf e}_2, {\bf e}_3$$ centered at $$O$$ such that the material points $$Q$$ of body $$B$$ have constant (in time) position vectors when refereed to that frame:

$$Q-O = \sum_{k=1}^e x_k(Q) {\bf e}_k$$, independently of the motion of $$B$$.

Now consider another reference frame $$O'$$, $${\bf e}'_1, {\bf e}'_2, {\bf e}'_3$$, where the body appears to move.

If you know the position in thaty space of $$O$$ and $${\bf e}_1, {\bf e}_2, {\bf e}_3$$, you completely know the position of all points of $$B$$.

To describe the motion of the body in the space of $$O'$$, $${\bf e}'_1, {\bf e}'_2, {\bf e}'_3$$ we can always proceed as follows.

1. Assign the translatory motion of $$O$$: $$O= O' + \sum_{j=1}^{3}x_j(t) {\bf e}'_j\:.$$

2. Define a third triple of axes $${\bf e}''_1, {\bf e}''_2, {\bf e}''_3$$ centered on $$O$$ and always parallel to $${\bf e}'_1, {\bf e}'_2, {\bf e}'_3$$.

3. Describe how the axes $${\bf e}_1(t), {\bf e}_2(t), {\bf e}_3(t)$$ (at rest with $$B$$) are seen in the triple in 2. This just amounts to assigning a map $$R=R(t) \in O(3)$$ such that $${\bf e}_j(t)= \sum_{k=1}^3 R_{jk}(t){\bf e}''_k$$ where every $$R(t)$$ is a rotation matrix.

It is clear that all the procedure can be performed for every choice of the point $$O\in B$$. Not only, you can also choose a point $$O_1$$ which is at rest in the space defined by $$O$$, $${\bf e}_1, {\bf e}_2, {\bf e}_3$$ instead of $$O$$ itself. It does not matter if $$O_1$$ is a material point of $$B$$. It is sufficient that it is at rest with $$B$$.