Weak convergence and averaging for ODE. (June 2016)

• Record Type: Journal Article
• Authors: Evans, Lawrence C.; Zhang, Te
• Is Part Of: Nonlinear analysis. Volume 138(2016)
• Page Start: 83
• Page End: 92
• Publication Date: 2016-06
• Subjects: Weak convergence -- Stabilization -- Adiabatic invariance; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.10.011
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: This mostly expository paper shows how weak convergence methods provide simple, elegant proofs of (i) the stabilization of an inverted pendulum under fast vertical oscillations, (ii) the existence of particle traps induced by rapidly varying electric fields and (iii) the adiabatic invariance of ∫ Γ p d x for slowing varying planar Hamiltonian dynamics. Under an appropriate, but very restrictive, unique ergodicity assumption, the proof of (iii) extends also to many degrees of freedom.