Abstract
The time-scales theory provides a powerful theoretical tool for studying differential and difference equations simultaneously. With regard to Herglotz type variational principle, this generalized variational principle can deal with non-conservative or dissipative problems. Combining the two tools, this paper aims to study time-scales Herglotz type Noether theorem for delta derivatives of Birkhoffian systems. We introduce the time-scales Herglotz type variational problem of Birkhoffian systems firstly and give the form of time-scales Pfaff–Herglotz action for delta derivatives. Then, time-scales Herglotz type Birkhoff’s equations for delta derivatives are derived by calculating the variation of the action. Furthermore, time-scales Herglotz type Noether symmetry for delta derivatives of Birkhoffian systems are defined. According to this definition, time-scales Herglotz type Noether identity and Noether theorem for delta derivatives of Birkhoffian systems are proposed and proved, which can become the ones for delta derivatives of Hamiltonian systems or Lagrangian systems in some special cases. Therefore, it is shown that the results of Birkhoffian formalism are more universal than Hamiltonian or Lagrangian formalism. Finally, the time-scales damped oscillator and a non-Hamiltonian Birkhoffian system are given to exemplify the superiority of the results.
Highlights
In 1988, Hilger proposed the definition of a time scale T, which is an arbitrary non-empty closed subset of the real numbers R, in order to analyse continuous and discrete systems uniformly [1]
It is well known that Noether theorem reveals that conservation quantities of mechanics are directly related to the invariance of actions under infinitesimal transformations
The study of time-scales symmetries and conservation quantities was extended to Birkhoffian systems [11] and Hamiltonian systems [12,13,14]
Summary
In 1988, Hilger proposed the definition of a time scale T, which is an arbitrary non-empty closed subset of the real numbers R, in order to analyse continuous and discrete systems uniformly [1]. Based on the Herglotz variational problem, Lagrangians and Hamiltonians with physical meaning can be established for non-conservative systems. They constructed Lagrangians of Herglotz type with physical meaning, such as vibrating string under viscous forces, non-conservative electromagnetic theory, non-conservative Schrödinger equation and Klein–Gordon equation, to describe non-conservative systems and quantum systems [18] 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. The time-scales Herglotz type Noether theorem of Birkhoffian systems has important practical applications for conservative and non-conservative processes in continuous and discrete cases, for example, finding a new solution from a known one, reducing equations, testing computer code and so on.
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