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

Abstract

The variational problem of Herglotz type and Noether's theorem for a time-delayed Hamiltonian system are studied. Firstly, the variational problem of Herglotz type with time delay in phase space is proposed, and the Hamilton canonical equations with time delay based on the Herglotz variational problem are derived. Secondly, by using the relationship between the non-isochronal variation and the isochronal variation, two basic formulae of variation of the Hamilton–Herglotz action with time delay in phase space are derived. Thirdly, the definition and criterion of the Noether symmetry for the time-delayed Hamiltonian system are established and the corresponding Noether's theorem is presented and proved. The theorem we obtained contains Noether's theorem of a time-delayed Hamiltonian system based on the classical variational problem and Noether's theorem of a Hamiltonian system based on the variational problem of Herglotz type as its special cases. At the end of the paper, an example is given to illustrate the application of the results.

Herglotz’s Variational Problem for Non-Conservative System with Delayed Arguments under Lagrangian Framework and Its Noether’s Theorem

“…For the Hamilton system with delayed arguments, the functional z can be defined by the differential equation [30]…”

“…In Reference [30], Herglotz type Noether's theorem for the Hamilton system with delayed arguments was studied. However, the above Equations ( 42) and (45) were not obtained in [30] due to an error in calculating the non-isochronous variation in the interval t ∈ [t 0 − τ, t 0 ] .…”

“…Third, HGVP unifies conservative and non-conservative processes into the same dynamics model, and thus can systematically deal with the actual dynamical problems. Noether's theorems [23,24] based on HGVP were extended to fractional order models [25][26][27][28][29], non-conservative Hamilton systems [30][31][32], non-holonomic systems [33], Birkhoff systems [34][35][36][37], non-conservative classical and quantum systems [38][39][40], and adiabatic invariants [41,42], etc. Although some advances have been made in the study of HGVP and Noether's theorems, but little work has been done on the HGVP with time delay and its symmetry and conservation laws.…”

A new type of adiabatic invariants for disturbed non-conservative nonholonomic system*

“…is the m-th order adiabatic invariant of the non-conservative nonholonomic system. Proof According to condition (12) and Eq. ( 7), we have…”

“…[5] Further, according to the Herglotz variational principle, Zhang studied the Noether's theorem and conserved quantities in phase space, [6] for non-conservative nonholonomic system, [7] for Birkhoffian system, [8][9][10] and with time delay. [11,12] Recently, many results have been obtained about the fractional Herglotz variational principle. [13][14][15][16][17] The study of symmetry and invariants of dynamic systems is of great significance.…”