One establishes some convexity criteria for sets in $R^2$. They will be applied in a further Note to treat the existence of solutions to minimum time problems for certain Lagrangian systems referred to two coordinates, one of which is used as a control. These problems regard the swing or the ski.

This Note is the Part II of a previous Note with the same title. One refers to holonomic systems $\mathrm{\Sigma }=\mathcal{A}\bigcup \mathcal{U}$ with two degrees of freedom, where the part $\mathcal{A}$ can schemetize a swing or a pair of skis and $\mathcal{U}$ schemetizes whom uses $\mathcal{A}$. The behaviour of $\mathcal{U}$ is characterized by a coordinate used as a control. Frictions and air resistance are neglected. One considers on $\mathrm{\Sigma }$ minimum time problems and one is interested in the existence of solutions. To this aim one determines a certain structural condition $\mathrm{\Gamma }$ which implies...

In the present work, divided in three parts, one considers a real skis-skier system, ${\mathrm{\Sigma }}_R$, descending along a straight-line $l$ with constant dry friction; and one schematizes it by a holonomic system $\mathrm{\Sigma }=A\cup U$, having any number $n\ge 4$ of degrees of freedom and subjected to (non-ideal) constraints, partly one-sided. Thus, e.g., jumps and also «steps made with sliding skis» can be schematized by $\mathrm{\Sigma }$. Among the $n$ Lagrangian coordinates for $\mathrm{\Sigma }$ two are the Cartesian coordinates $\xi$ and $\eta$ of its center of mass, $C$, relative...