Conservation laws for invariant functionals containing compositions

• Published in: Applicable analysis Vol. 86; no. 9; pp. 1117 - 1126
• Main Authors: Frederico, GastÃo S. F., Torres, Delfim F. M.
• Format: Journal Article
• Language: English
• Published: Taylor & Francis Group 01.09.2007
• Subjects: AMS Subject Classifications: 49K05; DuBois-Reymond condition; Functionals containing compositions; Noether's theorem; Symmetries; Variational calculus
• ISSN: 0003-6811; 1563-504X
• DOI: 10.1080/00036810701584583

The study of problems of the calculus of variations with compositions is a quite recent subject with origin in dynamical systems governed by chaotic maps. Available results are reduced to a generalized Euler-Lagrange equation that contains a new term involving inverse images of the minimizing trajectories. In this work, we prove a generalization of the necessary optimality condition of DuBois-Reymond for variational problems with compositions. With the help of the new obtained condition, a Noether-type theorem is proved. An application of our main result is given to a problem appearing in the chaotic setting when one consider maps that are ergodic. §Presented orally at the 7th IFAC Symposium on Nonlinear Control Systems (NOLCOS 2007), in Pretoria, South Africa, 22-24 August, 2007.