Given a Lagrangian system with non-holonomic constraints we construct an almost product structure on the tangent bundle of the configuration manifold such that the projection of the Euler-Lagrange vector field gives the dynamics of the system. In a degenerate case, we develop a constraint algorithm which determines a final constraint submanifold where a completely consistent dynamics of the initial system exists.

We associate to a dynamic equation $\xi$ three different connections and then we consider the meaning of the vanishing of their curvatures. Some peculiarities of the case of autonomous dynamic equation polynomial in the velocities $\dot{q}$ are pointed out. Finally, using the so-called Helmholtz conditions, we investigate a particular example.

The homogeneity properties of two different families of geometric objects playing a crutial role in the non-autonomous first-order dynamics - semisprays and dynamical connections on $R\times TM$ - are studied. A natural correspondence between sprays and a special class of homogeneous connections is presented.

In some preceding works we consider a class $\mathcal{O}\mathcal{P}$ of Boltz optimization problems for Lagrangian mechanical systems, where it is relevant a line $l=l_{\gamma (\cdot )}$, regarded as determined by its (variable) curvature function $\gamma (\cdot )$ of domain $\left[s_0,s_1\right]$. Assume that the problem $\tilde{\mathcal{P}}\in \mathcal{O}\mathcal{P}$ is regular but has an impulsive monotone character in the sense that near each of some points ${\delta }_1$ to $\delta {}_{\nu }\gamma (\cdot )$ is monotone and $\left|\gamma {}^{\prime }\right(\cdot \left)\right|$ is very large. In [10] we propose a procedure belonging to the theory of impulsive controls, in order to simplify $\tilde{\mathcal{P}}$ into a structurally...

Dynamical properties of singular Lagrangian systems differ from those of classical Lagrangians of the form $L=T-V$. Even less is known about symmetries and conservation laws of such Lagrangians and of their corresponding actions. In this article we study symmetries and conservation laws of a concrete singular Lagrangian system interesting in physics. We solve the problem of determining all point symmetries of the Lagrangian and of its Euler-Lagrange form, i.e. of the action. It is known that every point...