Conservation laws for invariant functionals containing compositions

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.

💡 Research Summary

The paper addresses a recently emerging class of variational problems in which the functional to be minimized contains compositions of functions, i.e., terms of the form (f(g(x))) or (u(g(x))). Traditional calculus of variations deals with functionals that depend only on an independent variable (x), a dependent variable (u(x)), and its derivative (u’(x)). In that setting the Euler‑Lagrange equation and the Du Bois‑Reymond necessary condition provide the optimality criteria. When a composition appears, however, the classical theory is insufficient because the optimal trajectory is influenced by the inverse images of the inner map (g).

The authors first recall the generalized Euler‑Lagrange equation that has been proposed for such problems. This equation already contains an extra term involving a sum over the inverse image set (g^{-1}(x)). They make this term explicit and interpret it as a “force” exerted by the pre‑images on the trajectory.

The central contribution of the work is a new Du Bois‑Reymond condition adapted to functionals with compositions. Starting from a Lagrangian (L(x,u,u’,g(x))), they compute the total derivative with respect to (x) while applying the chain rule to the composed argument. The resulting condition reads

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