of the limit cycle $\cR$ is $\{(s,\dot s) : \dot s > -
  \nu(s)\}$.]{}

\[rem:shiriaev\] [Proposition \[prop:internal\_dynamics\_general\_properties\] has general implications. Consider a Lagrangian control system of the form  with configuration vector $q=
  (q_1,\ldots, q_n)$ and degree of underactuation one. Consider a VHC of the form $q_i = \Phi_i(q_n)$, $i=1,\ldots,n-1$, and suppose $q_n
  \in [\Re]_T$. For this system, it is shown in [@WesGriKod03; @ShiPerWit05; @MagCon13] that the reduced dynamics have the form , with $s$ replaced by $q_n$. In this context, Proposition \[prop:internal\_dynamics\_general\_properties\] gives sufficient conditions under which the reduced dynamics have an asymptotically stable closed orbit, implying that they are not Lagrangian. On the other hand, in Theorem 2 of [@Shiriaev2006900] (see also [@ShiPerWit05]) it is shown that for any $(s_0,\dot s_0)$, system  possesses an “integral of motion”[^2] $I(s,\dot s) = \dot s^2 +2 V(s)/ M(s) - \dot
  s_0^2/M(s)$, where $ M(s) = \exp \left\{ -2 \int_{s_0}^s
  \Psi_2(\tau) d \tau \right\}$, $V(s) = -\int_{s_0}^s \Psi_1(\mu)
  M(\mu) d\mu$. This integral of motion predicted by Theorem 2 of [@Shiriaev2006900] seemingly contradicts Proposition \[prop:internal\_dynamics\_general\_properties\]. Indeed, if $M(s)$ and $V(s)$ are well-defined functions, letting $L(s,\dot
  s) = (1/2) M(s) \dot s^2 - V(s)$ one has that $\frac{d}{dt}
  \frac{\partial L}{\partial \dot s} - \frac{\partial L}{\partial 