A new type of adiabatic invariant for fractional order non-conservative Lagrangian systems

Abstract

The Herglotz variational problem is also known as Herglotz generalized variational principle whose action functional is defined by differential equation. Unlike the classical variational principle, the Herglotz variational principle gives a variational description of a holonomic non-conservative system. The Herglotz variational principle can describe not only all physical processes that can be described by the classical variational principlen, but also the problems that the classical variational principle is not applicable for. If the Lagrangian or Hamiltonian does not depend on the action functional, the Herglotz variational principle reduces to the classical integral variational principle. In this work, in order to describe the dynamical behavior of complex non-conservative system more accurately, we extend the Herglotz variational principle to the fractional order model, and study the adiabatic invariant for fractional order non-conservative Lagrangian system. Firstly, based on the Herglotz variational problem, the differential variational principle of Herglotz type and the differential equations of motion of the fractional non-conservative Lagrangian system are derived. Secondly, according to the relationship between the isochronal variation and the nonisochronal variation, the transformation of invariance condition of Herglotz differential variational principle is established and the exact invariants of the system are derived. Thirdly, the effects of small perturbations on fractional non-conservative Lagrangian systems are studied, the conditions for the existence of adiabatic invariants for the Lagrangian systems of Herglotz type based on Caputo derivatives are established, and the adiabatic invariants of Herglotz type are obtained. In addition, the exact invariant and adiabatic invariant of fractional non-conservative Hamiltonian system can be obtained by Legendre transformation. When <inline-formula><tex-math id="M1">\begin{document}$ \alpha \to 1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="22-20200488_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="22-20200488_M1.png"/></alternatives></inline-formula>, the Herglotz differential variational principle for fractional non-conservative Lagrangian system degrades into classical Herglotz differential variational principle, and the corresponding exact invariants and adiabatic invariants also degenerate into the classical exact invariants and adiabatic invariants of Herglotz type. At the end of the paper, the fractional order damped oscillator of Herglotz type is discussed as an example to demonstrate the results.