Fractional Time-Scales Noether’s Theorem for Non-Standard Birkhoffian System

In this work, Noether symmetries and conserved quantities of a non-standard Birkhoffian system based on the Caputo <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Δ</mo></semantics></math></inline-formula> Pfaff–Birkhoff principle on time scales are studied. Firstly, equations of motion for Caputo <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Δ</mo></semantics></math></inline-formula> non-standard Birkhoffian systems are set up from Caputo <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Δ</mo></semantics></math></inline-formula> variational principle. Secondly, invariance of Caputo non-standard Pfaff action on time scales is demonstrated, thus giving rise to Noether symmetry criterions which establish Noether’s theorems for the corresponding system. The validity of the methods and results presented in the paper is illustrated by means of examples provided at the end of the article.