Skew critical problems

Abstract

Skew critical problems occur in continuous and discrete nonholonomic Lagrangian systems. They are analogues of constrained optimization problems, where the objective is differentiated in directions given by an apriori distribution, instead of tangent directions to the constraint. We show semiglobal existence and uniqueness for nondegenerate skew critical problems, and show that the solutions of two skew critical problems have the same contact as the problems themselves. Also, we develop some infrastructure that is necessary to compute with contact order geometrically, directly on manifolds

Error analysis of variational integrators of unconstrained Lagrangian systems

“…Thus there is a manifold of nondegenerate critical points parametrized by z q ∈ T Q. Semiglobal persistence of these critical points follows by Theorem 2 of [4] i.e. there are neighborhoodsÛ…”

“…As has been stated, the central issue is a decrease in the order of accuracy, essentially due to a division by h. Proposition 4.4, which is yet another result of [4], tracks this in the context of Proposition 3.6. (M, h M ) and N be a manifolds, let π : E → N a vector bundle, and suppose f i andf i are as in Proposition 3.6, with k ≥ r .…”

“…The technical statements in the Theorem to this effect are immediate from Proposition 5 of [4], applied to the map π 23 •γ , where π 23 (h, v,ṽ) = (v,ṽ). …”

“…Proposition 6 of [4] implies maps agree to the same order as their graphs, and one order higher if their residuals are symmetric.…”

“…We extensively use the development in [4]. In this paper, all manifolds are assumed paracompact and Hausdorff.…”