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I run across the following Lagrangian, $$\mathcal{L} = m \dot{x}\dot{y} + \frac{1}{2} \gamma (x \dot{y} - \dot{x} y) $$

I can see how a variation with respect to $$ x, y $$ yields the (viscous) equations of motions

$$ \ddot{x} + \dot{x} = 0 \quad, \quad \ddot{y} - \dot{y} = 0 $$.

In the paper I attach this Lagrangian is described as being the physical Lagrangian for two coupled oscillators, one with negative and the other with positive friction (hence one is exponentially stable, while the other is not, as the equation of motion stress).

I do not understand how the lagrangian $$ \mathcal{L}$$ can correspond to two such oscillators.

Moreover, the whole idea of the formalism seems to conserve energy, in the sense that the (non-dampening) oscillator will absorb all the energy dissipated by the viscous oscillator.

But then the two speeds should be in magnitude equal, and not like $$ e^{t}$$ and $$e^{-t}$$.

I struggle to understand the physical picture, I wounder if anybody could help.

Thanks you very much

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# Dissipative Lagrangian

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