The configuration space of a system of particles $(m_i,x_i)$, $i=1,\dots,n$, subject to constraints $$\Phi (x)=0,\qquad \Phi\colon \mathbb R^{3n}\to \mathbb R ^{3n-k},\qquad x=(x_1,...,x_n),$$
if the constraint is nice enough (i.e. if $0$ is a regular value of $\Phi$), can be described as a $k$-dimensional submanifold of $\mathbb R ^{3n}$, and this clearly has some advantages. Those systems are called “autonomous” (and if someone could throw some light on the terminology I'd also be grate).

Now, suppose that $\Phi$ depends on time, i.e. the constraints are:$$\Phi(x,t)=0.$$
In this case it isn't immediately obvious to me what would be the most natural way to describe the configuration space. For example, one might define $g^t(x)=\Phi(x,t)$, suppose that $0\in \mathbb R ^{3n-k}$ is still a regular value of $g^t$ and define the configuration space at time $t$ as $$M^t=(g^t)^ {-1}(0),$$
and describe the position of the system at time $t$ with $k+1$ parameters $(q_1,\dots,q_k,t)$. This works good if, for example, the manifolds $M^t$ are essentially the same: for example, if $M^t\subset \mathbb R ^3$ is a ring that rotates about the $z$ axis, the manifold is simply $S^1$. But is this description always possible? I mean, $M^t$ could possibly change in time so that the coordinates $(q_1,\dots,q_k,t)$ don't mean a thing for some $t$.

Another conceivable way, I suppose, would be to consider the ($k+1$-dimensional) manifold: $$M=\Phi ^{-1}(0)\subset \mathbb R ^{3n+1}\ni (x_1,...,x_n,t).$$
For example, in this post, I sketched a proof of the Noether's theorem for non-autonomous systems following a similar idea. The main problem that occurs to me is that, in this way, we make the statement of propositions like, for example, d'Alembert principle more complicated, because we can no more consider virtual displacements as tangent vectors to the configuration space.

**So, to summarize**: given a constraint of the form $\Phi(x,t)=0$, what is the most natural way to describe the configuration space of such a system?

This post imported from StackExchange Physics at 2014-04-25 01:58 (UCT), posted by SE-user pppqqq