My question is based on thoughts after reading to a specific section in the paper "On Contraction Analysis for Nonlinear Systems" by W. Lohmiller and JJ. Slotine, **Section 4.2 Constrained Systems**. Those interested in answering my question should probably read this section first.

**Question**

Suppose I have an $n$ dimensional nonlinear dynamical system $$x'(t)=f(x) \quad x\in \mathbb{R}^n $$ with converges to a $m$-dimensional submanifold $\mathcal{M}=\{x:g(x)=0\}$ for any initial condition ($\mathcal{M}$ is not invariant for all $t$, but all attracting invariant subsets and equilibrium points of the system are in $\mathcal{M}$. I am mentioning this specifically because the background I have I read on constrained systems assumes $\mathcal{M}$ is invariant which is not the case here). Can I rewrite the system in terms of $n-1$ dimensional ODE with a constraint, such that the ode system has the same dynamics, is of the form of Section 4.2? That is $$z'=\underbrace{f(z)}_{\text{constrained dynamics $z=g(x)$}}+\underbrace{n}_{\text{Superimposed flow normal to the manifold}} $$

**Background to my question**

After reading a review on numerical methods for Lypunov functions and "On Contraction Analysis for Nonlinear Systems" I am in love with using contraction metrics for analysis of dynamical systems as it helps me answer some complex problems (none are related directly to control theory). I have a system that has the properties specified in the question. If I could apply the results as seen in Section 4.2 to my system (separating the flow projected onto a manifold and flow normal to a manifold) that would be great.

**Notes:**

- If you need any clarification please comment.
- If it is possible to separate a dynamical system in terms of flow normal to a manifold and flow projected onto a manifold that would be acceptable as well, as that is essentially what I am looking for.
- I am not familiar with constrained dynamical systems. I may be missing something quite obvious. Feel free to tell me, so I can update my post as necessary.
- If there are publications that specifically expand/apply the results of Section 4.2 I would greatly appreciate any citations.
- Feel free to edit the tags of my question. I have put in the control tag given the material I have read and the rest of the content of the paper I thought it was appropriate.