Abstract
Because Herglotz’s variational problem achieves the variational representation of non-conservative dynamic processes, its research has attracted wide attention. The aim of this paper is to explore Herglotz’s variational problem for a non-conservative system with delayed arguments under Lagrangian framework and its Noether’s theorem. Firstly, we derive the non-isochronous variation formulas of Hamilton–Herglotz action containing delayed arguments. Secondly, for the Hamilton–Herglotz action case, we define the Noether symmetry and give the criterion of symmetry. Thirdly, we prove Herglotz type Noether’s theorem for non-conservative system with delayed arguments. As a generalization, Birkhoff’s version and Hamilton’s version for Herglotz type Noether’s theorems are presented. To illustrate the application of our Noether’s theorems, we give two examples of damped oscillators.
Highlights
Time delay is a common phenomenon in nature and engineering
Time delays have often been ignored in the past and many problems have been solved, with the increasingly precise requirements for the dynamical behavior and control of complex systems, the effects of time delays on the system need to be considered
In 2013, in reference [8], we extended the results of [7] in three aspects: from Lagrange system to general non-conservative system; from a group of point transformations corresponding to generalized coordinates and time to a group of transformations that depend on generalized velocities; from Noether symmetry to Noether quasi-symmetry
Summary
Time delay is a common phenomenon in nature and engineering. time delays have often been ignored in the past and many problems have been solved, with the increasingly precise requirements for the dynamical behavior and control of complex systems, the effects of time delays on the system need to be considered. Noether’s theorems with time delay have been extended to high-order variational problems [9], fractional systems [10], Hamilton systems [11], nonholonomic systems [12], Birkhoff systems [13,14], and dynamics on time scales [15,16], etc. Different from the classical variational principle (CVP), the advantages of HGVP are as follows It achieves a variational representation of the process of non-conservative dynamics. Based on the two aspects as stated above, our motivation is to apply HGVP to the time-delay mechanical system and study Herglotz’s variational problem for a non-conservative system with delayed arguments under Lagrangian framework and its Noether’s theorem. Herglotz type Noether’s theorem for non-conservative systems with delayed arguments is proved.
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