Conservation laws for Birkhoffian systems of Herglotz type

Abstract

Conservation laws for the Birkhoffian system and the constrained Birkhoffian system of Herglotz type are studied. We propose a new differential variational principle, called the Pfaff-Birkhoff-d'Alembert principle of Herglotz type. Birkhoff's equations for both the Birkhoffian system and the constrained Birkhoffian system of Herglotz type are obtained. According to the relationship between the isochronal variation and the nonisochronal variation, the conditions of the invariance for the Pfaff-Birkhoff-d'Alembert principle of Herglotz type are given. Then, the conserved quantities for the Birkhoffian system and the constrained Birkhoffian system of Herglotz type are deduced. Furthermore, the inverse theorems of the conservation theorems are also established.

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

“…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*

“…[2,3] Torres et al studied the higher-order Herglotz variational problem [4] and Herglotz variational problem with time delay and its Noether theorem. [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.…”

Time-scales Herglotz type Noether theorem for delta derivatives of Birkhoffian systems

“…In the majority of above articles, their variational principles are the classical extremum principles, for example, the famous Hamilton principle, whose action is defined by an integral. However, in general, the Hamilton principle of non-conservative systems is an instability action principle, because the absence of a functional makes its variation equal to zero [15]. Whereas, Herglotz type variational principle can deal with this problem to give the variational description for non-conservative systems by the action functional defined by a differential equation [16].…”

“…Moreover, when these functions do not depend on the action functional, Herglotz variational principle can be reduced to the classical integral variational principle, which can deal with conservative problems. Since Herglotz type variational principle provides a new method for studying non-conservative systems, Herglotz type Noether theorems of mechanical systems have been investigated in recent decades, including non-conservative Lagrangian systems [19,20], non-conservative Hamiltonian systems [21], Birkhoffian systems [15,22], non-conservative non-holonomic systems [23] and other complex systems [24–31]. But so far, time-scales Herglotz variational principle is rarely studied, and the results are limited to Lagrangian formalism [32,33] and Hamiltonian formalism [34].…”