We see how the first jet bundle of curves into affine space can be realized as a homogeneous space of the Galilean group. Cartan connections with this model are precisely the geometric structure of second-order ordinary differential equations under time-preserving transformations - sometimes called KCC-theory. With certain regularity conditions, we show that any such Cartan connection induces “laboratory” coordinate systems, and the geodesic equations in this coordinates form a system of second-order...

Some geometric objects of higher order concerning extensions, semi-sprays, connections and Lagrange metrics are constructed using an anchored vector bundle.

Let (P,ω) be a symplectic manifold. We find an integrability condition for an implicit differential system D' which is formed by a Lagrangian submanifold in the canonical symplectic tangent bundle (TP,ὡ).

Let $\mathrm{\Sigma }$ be a constrained mechanical system locally referred to state coordinates $(q^1,\mathrm{\dots },q^N,{\gamma }^1,\mathrm{\dots },{\gamma }^M)$. Let $({\tilde{\gamma }}^1\mathrm{\dots }{\tilde{\gamma }}^M)(\cdot )$ be an assigned trajectory for the coordinates ${\gamma }^{\alpha }$ and let $u(\cdot )$ be a scalar function of the time, to be thought as a control. In [4] one considers the control system ${\mathrm{\Sigma }}_{\widehat{\gamma }}$, which is parametrized by the coordinates $(q^1,\mathrm{\dots },q^N)$ and is obtained from $\mathrm{\Sigma }$ by adding the time-dependent, holonomic constraints ${\gamma }^{\alpha }={\widehat{\gamma }}^{\alpha }(t):={\tilde{\gamma }}^{\alpha }(u(t))$. More generally, one can consider a vector-valued control $u(\cdot )=(u^1,\mathrm{\dots },u^M)(\cdot )$ which is directly identified with $\widehat{\gamma }(\cdot )=({\widehat{\gamma }}^1,\mathrm{\dots },{\widehat{\gamma }}^M)(\cdot )$. If one denotes the momenta conjugate...

We give different notions of Liouville forms, generalized Liouville forms and vertical Liouville forms with respect to a locally trivial fibration π:E → M. These notions are linked with those of semi-basic forms and vertical forms. We study the infinitesimal automorphisms of these forms; we also investigate the relations with momentum maps.

We discuss Lyapunov stability/instability of both lower and upper equilibria of free damped pendulum with periodically oscillating suspension point. We recall the results of Bogolyubov and Kapitza, provide new effective criteria of stability/instability of the equilibria of pendulum equation, and give the exact and complete proofs. The criteria obtained are formulated in terms of positivity/negativity of Green's functions of the periodic boundary value problems for linearized equations. Furthermore,...

The system of zero-pressure gas dynamics conservation laws describes the dynamics of free particles sticking under collision while mass and momentum are conserved. The existence of such solutions was established some time ago. Here we report a uniqueness result that uses the Oleinik entropy condition and a cohesion condition. Both of these conditions are automatically satisfied by solutions obtained in previous existence results. Important tools in the proof of uniqueness are regularizations, generalized...

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...