American Institute of Mathematical Sciences

In this paper, we consider a generalization of variational calculus which allows us to consider in the same framework different cases of mechanical systems, for instance, Lagrangian mechanics, Hamiltonian mechanics, systems subjected to constraints, optimal control theory... This generalized variational calculus is based on two main notions: the tangent lift of curves and the notion of complete lift of a vector field. Both concepts are also adapted for the case of skew-symmetric algebroids, therefore, our formalism easily extends to the case of Lie algebroids and nonholonomic systems (see also [20]). Hence, this framework automatically includes reduced mechanical systems subjected or not to constraints. Finally, we show that our formalism can be used to tackle the case of discrete mechanics, including reduced systems, systems subjected to constraints and discrete optimal control theory.

An inverted planar pendulum with horizontally moving pivot point is considered. It is assumed that the law of motion of the pivot point is given and the pendulum is moving in the presence of dry friction. Sufficient conditions for the existence of solutions along which the pendulum never falls below the horizon are presented. The proof is based on the fact that solutions of the corresponding differential inclusion are right-unique and continuously depend on initial conditions, which is also shown in the paper.

In this paper we give the Bohr-Sommerfeld-Heisenberg quantization of the mathematical pendulum.

We consider a regular distribution \begin{document}$\mathcal{D}$\end{document} in a Riemannian manifold \begin{document}$(M, g)$\end{document}. The Levi-Civita connection on \begin{document}$(M, g)$\end{document} together with the orthogonal projection allow to endow the space of sections of \begin{document}$\mathcal{D}$\end{document} with a natural covariant derivative, the intrinsic connection. Hence we have two different covariant derivatives for sections of \begin{document}$\mathcal{D}$\end{document}, one directly with the connection in \begin{document}$(M, g)$\end{document} and the other one with this intrinsic connection. Their difference is the second fundamental form of \begin{document}$\mathcal{D}$\end{document} and we prove it is a significant tool to characterize the involutive and the totally geodesic distributions and to give a natural formulation of the equation of motion for mechanical systems with constraints. The two connections also give two different notions of curvature, curvature tensors and sectional curvatures, which are compared in this paper with the use of the second fundamental form.

This paper presents a coordinate-free formulation of virtual holonomic constraints for underactuated Lagrangian control systems on Riemannian manifolds. It is shown that when a virtual constraint enjoys a regularity property, the constrained dynamics are described by an affine connection dynamical system. The affine connection of the constrained system has an elegant relationship to the Riemannian connection of the original Lagrangian control system. Necessary and sufficient conditions are given for the constrained dynamics to be Lagrangian. A key condition is that the affine connection of the constrained dynamics be metrizable. Basic results on metrizability of affine connections are first reviewed, then employed in three examples in order of increasing complexity. The last example is a double pendulum on a cart with two different actuator configurations. For this control system, a virtual constraint is employed which confines the second pendulum to within the upper half-plane.