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I was reading the following article when I came across the interesting sentence

"non-Hamiltonian [symplectic group] actions also occur in physics"

I took a cursory look at the article cited but nothing jumped out at me to back up the claim. Does anyone have a good example of such a thing? Ideally a classical system, if possible, as I don't have much contact with the quantum world :)

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  • $\begingroup$ What exactly do you understand as a "Hamiltonian symplectic action"? A symplectic Lie group action which has a momentum map? $\endgroup$
    – Tobias Diez
    Commented Sep 27, 2018 at 13:44
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    $\begingroup$ Based on your reply to Carlo Beenakker, here's an example. Consider a symplectic torus $(\mathbb{T}^2, dx\wedge dy)$, with periodic coordinates $x$ and $y$. Then the vector field $\partial/\partial x$ (and its flow) is symplectic, but not Hamiltonian. Granted, this example is a bit artificial, but it's not hard to see how similar things might come up in the wild. In general, the obstruction for a symplectic vector field to be Hamiltonian is topological, living in $H^1$ of the symplectic manifold (which needs to be non-simply connected for some obstructions to exist). $\endgroup$
    – Igor Khavkine
    Commented Sep 27, 2018 at 13:56
  • $\begingroup$ Clearly, if you have a manifold where the symplectic structure is exact (eg a cotangent bundle $(T^*L, d\lambda)$), and if $X$ is a symplectic vector field leaving $\lambda$ invariant, then $ H_X = \lambda(X) $ is a Hamiltonian function, because $dH_X = d\iota_X \lambda = L_X \lambda - \iota_X d\lambda = - \iota_X \omega$. If $G$ is a compact Lie group acting, average first $\lambda$ over $G$ to obtain an invariant primitive for $\omega$ so that such an action is always Hamiltonian. You will either need to consider actions of non-compact groups or symplectic manifolds that are non-exact. $\endgroup$
    – Klaus Niederkrüger
    Commented Jun 14, 2022 at 18:42

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The master thesis Nonholonomic Dynamical Systems by Brett Ryland contains several examples of non-Hamiltonian systems from classical physics: the dynamics of a laser and the evolution of a gas flame (page 14), the rattleback (page 29).

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    $\begingroup$ Dear Carlo, thank you for your answer, but this is not quite what I am looking for. As far as I understand, these systems are also non-symplectic (of which there are, of course, a great number). I am interested specifically in the applications of non-hamiltonian symplectic transformations $\endgroup$
    – R Mary
    Commented Sep 27, 2018 at 13:12

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