s} =
  0$, and hence the reduced dynamics  are Lagrangian. This fact rules out the existence of isolated closed orbits for  predicted by Proposition \[prop:internal\_dynamics\_general\_properties\]. This contradiction is due to the fact that Theorem 2 in [@Shiriaev2006900] implicitly assumes that the functions $M(s)$ and $V(s)$ are well-defined. This is always true if $s \in
  \Re$. However, when $s \in [\Re]_T$, the functions $M(s)$ and $V(s)$ may be multi-valued, in which case the Lagrangian $L(s,\dot s)$ is undefined. Indeed, the assumptions of Proposition \[prop:internal\_dynamics\_general\_properties\] imply that both $M(s)$ and $V(s)$ are multi-valued. We refer the reader to Section IV of [@MagCon13] for a detailed discussion on this subject, and to [@MohMagCon13] for necessary and sufficient conditions under which  is Lagrangian.]{}

\[rem:Lagrange\] [In Lagrangian mechanics, the Lagrange-d’Alembert principle implies that the enforcement of an ideal holonomic constraint (i.e., a constraint for which the constraint forces do not produce work) on a Lagrangian system gives rise to a reduced Lagrangian system. Proposition \[prop:internal\_dynamics\_general\_properties\] shows that this result is not true when enforcing VHCs on Lagrangian control systems. There is, therefore, a sharp difference between [*ideal*]{} and [*virtual*]{} holonomic constraints when it comes to the reduced motion they induce. This difference is due to 