# Conservation laws for invariant functionals containing compositions

## TL;DR

This paper generalizes the calculus of variations to include compositions, deriving new optimality conditions and a Noether-type theorem, with applications to chaotic and ergodic dynamical systems.

## Contribution

It introduces a generalized necessary optimality condition and a Noether-type theorem for variational problems involving compositions, expanding the theoretical framework.

## Key findings

- Derived a generalized Euler-Lagrange equation with inverse image terms
- Proved a Noether-type symmetry theorem for composition-based variational problems
- Applied results to chaotic and ergodic dynamical systems

## Abstract

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.