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I recently found out that it is possible to formulate a Hamiltonian for a system with time-dependent coordinates such that the Hamiltonian is not the same as the energy When is the Hamiltonian of a system not equal to its total energy? and that has me wondering if it is possible to formulate a Hamiltonian for a damped system under these conditions. I know that Hamilton's equations require that energy be conserved, but if the coordinates are time-dependent, would it still be possible to formulate and solve the problem?

I started trying to do it for a damped simple harmonic oscillator by starting with the Lagrangian for the system

$$L=e^{\gamma * t}*(\frac{mv^2}{2}-\frac{kx^2}{2}),$$

but I keep on coming up with a Hamiltonian that is just equal to the energy

$$H=e^{\gamma * t}*(\frac{mv^2}{2}+\frac{kx^2}{2}).$$

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Alexandra Ellis
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The following might help:

$H = \frac{1}{2}(mv^2 + kx^2) + \gamma mkvx$

decays exponentially with time along the solution of the damped system. Check by differentiating $H$ with respect to $t$ and using the equations of the system. So the "energy" $H$ decays exponentially instead of remaining constant.

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user30850
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