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For questions involving the Lagrangian formulation of a dynamical system. Namely, the application of an action principle to a suitably chosen Lagrangian or Lagrangian Density in order to obtain the equations of motion of the system.

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Caldeira-Leggett Dissipation: frequency shift due to bath coupling

To expand on Adam's answer, let's see the effect of a static $Q$ on the system without the $\Delta \Omega^2$. The Lagrangian becomes $$L = -\frac{1}{2}\Omega^2Q^2 - Q\sum_i f_iq_i + \sum_i \frac{1}{2} …
mthibodeau's user avatar