# How come a rigid body has 6 degrees of freedoms (DOFs) ? Isn't velocity a DOF?

For rigid body we need to know position of three points and their velocities to determine everything. So that would make 12 DOF. Why do text books say it has six DOFs?

## 1 Answer

A rigid body has 6 configuration degrees of freedom because its most general configuration can be obtained by translating (3 degrees of freedom) and rotating (3 degrees of freedom) its initial configuration. A mathy way of saying this is that its configuration manifold is $\mathbb R^3\times \mathrm{SO}(3)$.

However, you are right that the phase space of a rigid body is 12-dimensional because each independent configuration degree of freedom corresponds to an independent momentum degree of freedom.

• DOF is a vague term at best and should just not be used. – AHusain Aug 8 '16 at 15:24
• @AHusain A few years ago, I might have agreed with you, but I've come to realize that sometimes vague terms are fine (and even useful) to use in the appropriate context. In particular, in the context of classical mechanics, it's usually clear from the context that the degrees of freedom are the dimensions of a certain manifold (configuration or phase), and with this understanding, the vagueness is eliminated. Once this happens, and you know what you're talking about, the terminology is actually quite intuitively appealing in my opinion. – joshphysics Aug 8 '16 at 15:33