Conserved quantities and adiabatic invariants of fractional Birkhoffian system of Herglotz type**Project supported by the National Natural Science Foundation of China (Grant Nos. 11972241, 11572212, and 11272227), the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20191454), and the Innovation Program for Postgraduade in Higher Education Institutions of Jiangsu Province, China (Grant No. KYCX19 2013).

Abstract

In order to further study the dynamical behavior of nonconservative systems, we study the conserved quantities and the adiabatic invariants of fractional Brikhoffian systems with four kinds of different fractional derivatives based on Herglotz differential variational principle. Firstly, the conserved quantities of Herglotz type for the fractional Brikhoffian systems based on Riemann–Liouville derivatives and their existence conditions are established by using the fractional Pfaff–Birkhoff–d′Alembert principle of Herglotz type. Secondly, the effects of small perturbations on fractional Birkhoffian systems are studied, the conditions for the existence of adiabatic invariants for the Birkhoffian systems of Herglotz type based on Riemann–Liouville derivatives are established, and the adiabatic invariants of Herglotz type are obtained. Thirdly, the conserved quantities and adiabatic invariants for the fractional Birkhoffian systems of Herglotz type under other three kinds of fractional derivatives are established, namely Caputo derivative, Riesz–Riemann–Liouville derivative and Riesz–Caputo derivative. Finally, an example is given to illustrate the application of the results.