Noether theory for Birkhoffian systems with nabla derivatives

There are discrete phenomena which happen only on discrete time or hold discrete space structures such as economy series, population dynamics et al.. Then there is a tool needed for these discrete issues or applications. Time scale is one of the useful tools to solve some discrete problems. In this paper, time scale is used to establish discrete PfaffBirkhoff principle and achieve discrete Birkhoff equations, discrete Noether identity and discrete conserved quantity for the discrete Birkhoffian system. Firstly, Birkhoff equations, Noether identity and Noether theorem with nabla derivatives on time scales are investigated by using the isochronous variational principle. Secondly, some special cases, especially the discrete Birkhoffian system are discussed. Thirdly, another method, i.e., the duality principle is introduced for the Birkhoffian system on time scales. And finally, an example is given to illustrate the results and methods.