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.
Highlights
It is well known that variational principles play an important role in the fields of mechanics, physics and engineering, etc
We will extend the variational problem of Herglotz type to a time-delayed Hamiltonian system and study the Noether symmetry and the conserved quantity for the system based on the generalized variational principle of Herglotz type
For the Hamiltonian system based on the variational problem of Herglotz type, if the infinitesimal transformations (4.2) of r-parameter Lie group of transformations are the Noether symmetry transformations of the system (7.9), the system exists with r independent conserved quantities, which are where l(t)
Summary
It is well known that variational principles play an important role in the fields of mechanics, physics and engineering, etc. We will extend the variational problem of Herglotz type to a time-delayed Hamiltonian system and study the Noether symmetry and the conserved quantity for the system based on the generalized variational principle of Herglotz type. According to the generalized variational principle of Herglotz [1,2], the variational problem of Herglotz type with time delay can be defined as follows: Determine the trajectories qs(t), ps(t) to extremize the value z(t1) where the functional z is defined by the differential equation z_(t) 1⁄4 ps(t)q_s(t) þ ps(t À t)q_s(t À t) À H(t,qs(t),ps(t),qs(t À t),ps(t À t),z(t) ). The above variational problem can be referred to as the generalized variational principle for a time-delayed Hamiltonian system of Herglotz type. A mechanical system whose motion is described by equation (3.13) is called a time-delayed Hamiltonian system of Herglotz type
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