Discrete Lagrange-d'Alembert-Poincaré equations for Euler's disk
A nonholonomic connection \(\mathcal{A}\) on a principal \(G\)-bundle \(Q\) which is equipped with a \(G\)-invariant metric and a distribution \(\mathcal{D}\) such that \(TQ=\mathcal{D}+\mathcal{V}\) for the vertical distribution \(\mathcal{V}\) is characterized by the property that its horizontal tangent distribution is the orthogonal complement of the distribution \(\mathcal{S}:=\mathcal{D} \cap\mathcal{V}\) in \(\mathcal{D}\) at every point of \(Q\). Such a nonholonomic connection \(\mathcal{A}\) gives a vector bundle isomorphism \(\alpha _{\mathcal{A}}:\left( TQ\right) /G\rightarrow T\left( Q/G\right) \oplus\widetilde{\mathfrak{g}}\) for the adjoint bundle \(\widetilde{\mathfrak{g}}\) of the principal \(G\)-bundle, providing a reduced framework in which the variational Lagrange-d'Alembert principle can be formulated in a reduced form of Lagrange-d'Alembert-Poincaré equations. After some discretization process, this is applied to derive a system of reduced discrete equations for the study of Euler's disk, where a thick disk (or a thin cylinder) with a rotationally symmetric mass distribution with respect to its central axis is rolling without sliding on a rough surface with only one point of contact. The variational integrator obtained in the discretization of Lagrange-d'Alembert-Poincaré equations for this nonholonomic system is numerically simulated. The result shows that the contact point's path is closed and the energy level remains almost constant, in significant contrast with some decaying phenomena found in the simulation by the Runge-Kutta method.
discrete Lagrange-d'Alembert-Poincaré equations
scientific article; zbMATH DE number 6132863
scientific article