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I know this sounds rather insane, but it says so in my book. The argument is the following:

Given a damped harmonic oscilator $$\ddot{q}+\frac{b}{m}\dot{q}+\frac{k}{m}q=0 \tag1 $$

this system can be written in Hamiltonian form in the following way: $$H=\frac{p^2}{2m}e^{-bt/m}+\frac{kq^2}{2}e^{bt/m}$$ and using Hamilton's equations of motion it can be checked that this Hamiltonian really does give us the equation $(1)$

The argument why this is not dissipative is because of the Louville Theorem ,the system is not dissipative (so they claim) because it can be described by canonical formalism. That is, the solutions of this system are

$$q(t)=e^{-bt/2m}\Big(A\cos(\omega t)+B\sin(\omega t)\Big), \omega=\sqrt{k/m-b^ˇ/4m}$$ and $$p(t)=me^{bt/2m}\Big((B\omega-\frac{b}{2m}A)\cos(\omega t)-(A\omega+\frac{b}{2m}B)\sin(\omega t)\Big)$$

which would mean that as $q$ gets smaller, $p$ gets larger, and hence the area in phase space is conserved (as in the picture)Phase space for this problem (or at least it should be)

Since the area is proportional to $E/\omega$ and $\omega$ is a constant so is $E$!

This is obviously wrong, and there must be some hidden assumption we are not taking into account (and it probably has to to with the fact that $p$ is getting exponentially bigger with time..) What is this assumption, what is this trying to say to me, definitely not that Energy is constant in a damped oscillator, Right?

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  • $\begingroup$ Hmm, it is almost as if the dissipated energy is "hiding" in $p$ - could it be that this is a model where $p$ acts as an environment that absorbs dissipated energy. After all, a dampened harmonic oscillator in a box might be modelled as the oscillator driving a heat reservoir that doesn't produce any ordered reaction force. $\endgroup$ – Anders Sandberg Jun 10 '18 at 23:52
  • $\begingroup$ I think it may in the assumption that there is a hamiltonian for the equation of motion to begin with. Usually we would use the undamped oscillator Hamiltonian to get the equations of motion, and put in the damping by hand. Said otherwise, for a physical damped oscillator system you would say that the energy of the system is given by the usual harmonic hamiltonian, and not the one you wrote in the question $\endgroup$ – KF Gauss Jun 10 '18 at 23:56
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    $\begingroup$ Possible duplicates: physics.stackexchange.com/q/147341/2451 $\endgroup$ – Qmechanic Jun 11 '18 at 1:52
  • $\begingroup$ Hmm, could you give a reference for this? I was interested in reading what exactly is written in your textbook. $\endgroup$ – zorn Dec 9 '18 at 15:59
  • $\begingroup$ unfortunately, the textbook is not in English, but if you know Serbo-Croatian (or Russian) you can have a Look :) $\endgroup$ – Alexandar Solženjicin Dec 10 '18 at 16:51
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The Hamiltonian that you wrote is describing a complete dynamical system in which the energy is conserved and it is not suitable for dissipation systems, it could be a dynamical extension of the first law of thermodynamics. Your book's argument is true that "it can be described by canonical formalism". However, a dissipative system can be described by noncanonical Poisson brackets .

Professor P. J. Morrison has a nice article in which he reviewed and developed the bracket formulations of incomplete and complete dynamical systems.

http://w3fusion.ph.utexas.edu/ifs/ifsreports/IFSR%201337.pdf

I strongly recommend you to read this paper too:

http://iopscience.iop.org/article/10.1088/0031-8949/1982/T2B/032

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