Configuration space

Remarks

Example: a particle on a sphere of radius $\ell > 0$ has the single constraint $f(x) = \abs{x}^2 - \ell^2$ on $U = \R^3$. Its gradient $2x$ is nonzero on $Q$, so $Q$ is the sphere, a surface of dimension $3 - 1 = 2$. Non-example: the constraints $f_1(x) = z$ and $f_2(x) = z^2$ for one particle in $\R^3$ define the plane $z = 0$. Their gradients $(0,0,1)$ and $(0,0,2z)$ are dependent on that plane, and the count $3 - 2 = 1$ is wrong; independence excludes such families. For a system with no constraints, $Q$ is the open set $U$, for example $\R^{3N}$. More abstractly, a configuration space is any smooth manifold whose points label the positions of the system. For instance, $SO(3)$ is the configuration space of a rigid body with one fixed point.