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...

See Summary in Note I. First, on the basis of some results in [2] or [5]-such as Lemmas 8.1 and 10.1-the general (mathematical) theorems on controllizability proved in Note I are quickly applied to (mechanic) Lagrangian systems. Second, in case $\mathrm{\Sigma }$, $\chi$ and $M$ satisfy conditions (11.7) when $𝒬$ is a polynomial in $\dot{\gamma }$, conditions (C)-i.e. (11.8) and (11.7) with $𝒬\equiv 0$-are proved to be necessary for treating satisfactorily $\mathrm{\Sigma }$'s hyper-impulsive motions (in which positions can suffer first order discontinuities)....

In [1] I and II various equivalence theorems are proved; e.g. an ODE $(ℰ)\dot{z}=F(t,z,u,\dot{u})(\in {ℝ}^m)$ with a scalar control $u=u(\cdot )$ is linear w.r.t. $\dot{u}$ iff $(\alpha )$ its solution $z(u,\cdot )$ with given initial conditions (chosen arbitrarily) is continuous w.r.t. $u$ in a certain sense, or iff $(\beta )$

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...

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...

In applying control (or feedback) theory to (mechanic) Lagrangian systems, so far forces have been generally used as values of the control $u(\cdot )$. However these values are those of a Lagrangian co-ordinate in various interesting problems with a scalar control $u=u(\cdot )$, where this control is carried out physically by adding some frictionless constraints. This pushed the author to consider a typical Lagrangian system $\mathrm{\Sigma }$, referred to a system $\chi$ of Lagrangian co-ordinates, and to try and write some handy conditions,...

In variational calculus, the minimality of a given functional under arbitrary deformations with fixed end-points is established through an analysis of the so called second variation. In this paper, the argument is examined in the context of constrained variational calculus, assuming piecewise differentiable extremals, commonly referred to as extremaloids. The approach relies on the existence of a fully covariant representation of the second variation of the action functional, based on a family of...