Noether theory for Birkhoffian systems with nabla derivatives

Abstract

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 Pfaff-Birkhoff 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.

Lie Symmetry Theorem for Nonshifted Birkhoffian Systems on Time Scales

“…then corresponding symmetry is said the Lie symmetry for nonshifted constrained Birkhoffian system (13). Theorem 3 If ξ 0 and ξ μ satisfy the deterministic equation (30), and exist G(ta ρ ) satisfying the structural equation…”

“…Bartosiewicz and his co-workers [24] studied the time-scale second Euler-Lagrange equation, and derived Noether 􀆳 s conserved quantities on time scales by using this equation. Subsequently, many scholars [25][26][27][28][29][30][31][32][33][34] began to carry out a series of studies on time-scale the variational principle for constrained mechanical systems and Noether 􀆳 s theorem of different dynamic systems. However, on a time scale, there are few studies on Lie symmetry in mechanical systems.…”

Noether's symmetry and conserved quantity for a time-delayed Hamiltonian system of Herglotz type

“…Noether's theorem reveals the inherent relation between the Noether symmetry and the conserved quantity [16,17]. Recently, a series of important advances have been made in the research of Noether symmetry and conserved quantity [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]. In this section, we establish Noether's theorem for the time-delayed Hamiltonian system of Herglotz type.…”

Symmetries and perturbations of time-scale nonshifted singular systems

“…At present, there are many existing studies on the symmetries and perturbations of Lagrangian systems, [34][35][36][37][38][39] Hamiltonian systems, [40][41][42][43] and Birkhoffian systems [44][45][46][47][48][49] on the time scale. However, more research is needed on singular systems on the time scale.…”