On Lagrangian systems with some coordinates as controls

Rampazzo, Franco

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 82 (1988) no. 4, pp. 685-695

Résumé

Let $\Sigma$ be a constrained mechanical system locally referred to state coordinates $(q^{1},...,q^{N}, \gamma^{1},...,\gamma^{M})$. Let $(\tilde{\gamma}^{1}...\tilde{\gamma}^{M})(\cdot)$ 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 $\Sigma_{\hat{\gamma}}$, which is parametrized by the coordinates $(q^{1},...,q^{N})$ and is obtained from $\Sigma$ by adding the time-dependent, holonomic constraints $\gamma^{\alpha} = \hat{\gamma}^{\alpha}(t) := \tilde{\gamma}^{\alpha} (u(t))$. More generally, one can consider a vector-valued control $u(\cdot) = (u^{1},..., u^{M})(\cdot)$ which is directly identified with $\hat{\gamma}(\cdot) = (\hat{\gamma}^{1},..., \hat{\gamma}^{M})(\cdot)$. If one denotes the momenta conjugate to the coordinates $q^{i}$ by $p_{i}$, $i= 1,...,N$, it is physically interesting to examine the continuity properties of the input-output map $\phi : u(\cdot) \rightarrow (q^{i} ,p_{i})(\cdot)$ associated with the dynamical equations of $\Sigma_{\hat{\gamma}}$ with respect to e.g. the $C^{0}$ topologies on the spaces of the controls $u(\cdot)$ and of the solutions $(q^{i},p_{i})(\cdot)$. Furthermore, in the theory of hyperimpulsive motions (see [4]), even discontinuous control are implemented. Then it is crucial to investigate the continuity of $\phi$ also with respect to topologies that are weaker than the $C^{0}$ one. In order that the input-output map $\phi$ exhibits such continuity properties, the right-hand sides of the dynamical equation for $\Sigma_{\hat{\gamma}}$ have to be affine in the derivatives $\frac{d \hat{\gamma}^{1}} {dt},...,\frac{d\hat{\gamma}^{M}} {dt}$. If this is the case, the system of coordinates $(q^{i} ,\gamma^{\alpha})$ is said to be $M$-fit (for linearity). In this note we show that, in the case of forces which depend linearly on the velocity of $\Sigma$, the coordinate system $q^{i} ,\gamma^{\alpha})$ is $M$-fit if and only if certain coefficients in the expression of the kinetic energy are independent of the $q^{i}$. Moreover, if the forces are positional, for each $1$-fit coordinate system $(q^{\prime i} ,y^{\prime})$ there exists a reparametrization $(q^{j} ,\gamma)$ such that $\frac{\partial\gamma}{\partial q^{\prime i}} = 0$ holds for every $i = 1,...,N$ and the coordinates $(q^{i} ,\gamma)$ are locally geodesic.