Abstract:
Both conservative and nonconservative systems can be studied simultaneously using the differential variational principle of Herglotz type. For a perturbed system, in which parameters change with time, it is useful to find adiabatic invariants. Based on the Herglotz differential variational principle, we study the perturbation of infinitesimal transformations and adiabatic invariants for perturbed nonconservative Lagrangian systems. From the generalized Euler–Lagrange equation and the invariance condition for the Hamilton–Herglotz action under the group of infinitesimal transformations, we obtain an exact invariant of Herglotz type for a holonomic nonconservative system. We propose a definition of higher-order adiabatic invariants of Herglotz type and obtain such adiabatic invariants for nonconservative Lagrangian systems with small perturbations. We prove the corresponding inverse theorem of adiabatic invariants. As examples of using the obtained results, we consider an oscillator with square damping and a system with two degrees of freedom.
Innovation Program for Postgraduate in Higher Education Institutions of Jiangsu Province
KYZZ16_0479
This research was supported by the National Natural
Science Foundation of China (Grant No. 11572212) and the Innovation Program
for Postgraduate in Higher Education Institutions of Jiangsu Province
(KYZZ16_0479).
\Bibitem{TiaZha20}
\by Xue~Tian, Yi~Zhang
\paper Adiabatic invariants of Herglotz type for perturbed nonconservative Lagrangian systems
\jour TMF
\yr 2020
\vol 202
\issue 1
\pages 143--154
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\transl
\jour Theoret. and Math. Phys.
\yr 2020
\vol 202
\issue 1
\pages 126--135
\crossref{https://doi.org/10.1134/S0040577920010110}
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https://www.mathnet.ru/eng/tmf9723
https://doi.org/10.4213/tmf9723
https://www.mathnet.ru/eng/tmf/v202/i1/p143
This publication is cited in the following 3 articles:
J. Ryan, “When action is not least for systems with action-dependent Lagrangians”, Journal of Mathematical Physics, 64:3 (2023), 032901
Zhang Y., “Herglotz'S Variational Problem For Non-Conservative System With Delayed Arguments Under Lagrangian Framework and Its Noether'S Theorem”, Symmetry-Basel, 12:5 (2020)
Xu Xin-Xin, Zhang Yi, “A New Type of Adiabatic Invariant For Fractional Order Non-Conservative Lagrangian Systems”, Acta Phys. Sin., 69:22 (2020), 220401
Citation in format AMSBIB
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