Error analysis of forced discrete mechanical systems

Abstract

<p style='text-indent:20px;'>The purpose of this paper is to perform an error analysis of the variational integrators of mechanical systems subject to external forcing. Essentially, we prove that when a discretization of contact order <inline-formula><tex-math id="M1">\begin{document}$ r $\end{document}</tex-math></inline-formula> of the Lagrangian and force are used, the integrator has the same contact order. Our analysis is performed first for discrete forced mechanical systems defined over <inline-formula><tex-math id="M2">\begin{document}$ TQ $\end{document}</tex-math></inline-formula>, where we study the existence of flows, the construction and properties of discrete exact systems and the contact order of the flows (variational integrators) in terms of the contact order of the original systems. Then we use those results to derive the corresponding analysis for the analogous forced systems defined over <inline-formula><tex-math id="M3">\begin{document}$ Q\times Q $\end{document}</tex-math></inline-formula>.</p>

Exact discrete Lagrangian mechanics for nonholonomic mechanics

“…Therefore, a solution nh of ( 13) is of the form nh = L +λ a Z a , where Z a = (μ a i dq i ). The Lagrange multipliers may be computed by imposing the tangency condition (14), which is equivalent to This equation has a unique solution for the Lagrange multipliers if and only if the matrix C = (C ab ) = (Z a ( b )) is invertible at all points of D, which is equivalent to the compatibility condition (cf. [8]).…”

“…This is precisely the main idea behind geometric integration [3,18,33] and, in particular, of discrete mechanics and variational integrators [27]. In this last case, the construction of an exact discrete Lagrangian is a crucial element for the analysis of the error between the continuous trajectory and the numerical simulation derived by a variational integrator (see also [27,32] and [7,14] for forced systems). However, an open question is how to derive the exact discrete version for nonholonomic mechanics (see [29] for an attempt) and this is the main topic of the present paper.…”

Lagrangian reduction of forced discrete mechanical systems

“…Forced discrete mechanical systems have been known and studied for a number of years (see Part Three of [15]). Recently, the interest in these systems has been growing (see, for instance, [6,7,16,17]).…”

Erratum for "Error analysis of forced discrete mechanical systems"

“…hα + (h) ≥ 0 and hα − (h) ≤ 0 for all h ∈ (−a, a).We remark that the functions α + (h) := h and α − (h) := 0 for all h ∈ R didn't satisfy the original condition 2, but they meet this new condition and, so, can be part of a discretization of T Q data. It is relevant that this special case works because it is used in Examples 3.7 and 5.5, as well as Theorem 5.13 of [1]. All results of the paper remain unchanged for this new condition.…”

“…

Definition 3.1 in [1] is unnecessarily restrictive. In particular, its point 2 should be replaced by the slightly weaker condition:2.

…”