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This Note is the continuation of a previous paper with the same title. Here (Part II) we show that for every choice of the sequence $u{}_a(\cdot )$, ${\mathrm{\Sigma }}_a$'s trajectory $l_a$ after the instant $d+\eta a$ tends in a certain natural sense, as $a\to \mathrm{\infty }$, to a certain geodesic $l$ of $V_d$, with origin at $\left(\overline{q},\overline{u}\right)$. Incidentally $l$ is independent of the choice of applied forces in a neighbourhood of $\left(\overline{q},\overline{u}\right)$ arbitrarily prefixed.

In this Note (which will be followed by a second) we consider a Lagrangian system $\mathrm{\Sigma }$ (possibly without any Lagrangian function) referred to $N+1$ coordinates $q_1\mathrm{\cdots },q_N$, $u$, with $u$ to be used as a control, and precisely to add to $\mathrm{\Sigma }$ a frictionless constraint of the type $u=u\left(t\right)$. Let $\mathrm{\Sigma }$'s (frictionless) constraints be represented by the manifold $V_t$ generally moving in Hertz's space. We also consider an instant $d$ (to be used for certain limit discontinuity-properties), a point $\left(\overline{q},\overline{u}\right)$ of $V_d$, a value $\overline{p}$ for $\mathrm{\Sigma }$'s momentum conjugate...