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Adiabatic dynamics of one-dimensional classical Hamiltonian dissipative systems. (English) Zbl 1383.37061

Summary: A linearized plane pendulum with the slowly varying mass and length of string and the suspension point moving at a slowly varying speed is presented as an example of a simple 1D mechanical system described by the generalized harmonic oscillator equation, which is a basic model in discussion of the adiabatic dynamics and geometric phase. The expression for the pendulum geometric phase is obtained by three different methods. The pendulum is shown to be canonically equivalent to the damped harmonic oscillator. This supports the mathematical conclusion, not widely accepted in physical community, of no difference between the dissipative and Hamiltonian 1D systems.

MSC:

37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
81Q70 Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory
70H03 Lagrange’s equations
70H11 Adiabatic invariants for problems in Hamiltonian and Lagrangian mechanics
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