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Adiabatic invariants of two pendulums coupled by a spring

In this paper we use a didactic procedure to investigate the adiabatic invariants for the problem of two pendulums coupled by a spring, when one of the pendulums has its string pulled up slowly, in the limit of weak coupling. We verified that in this limit the normal frequencies are very close to the natural frequencies of the pendulums, as if they were uncoupled. In the case of adiabatic invariants, we observed that they are the ratios between the energies and the natural frequencies of the two pendulums.