5
$\begingroup$

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 :)

$\endgroup$
2
  • $\begingroup$ What exactly do you understand as a "Hamiltonian symplectic action"? A symplectic Lie group action which has a momentum map? $\endgroup$ – Tobias Diez Sep 27 '18 at 13:44
  • 4
    $\begingroup$ 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 Sep 27 '18 at 13:56
4
$\begingroup$

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).

$\endgroup$
1
  • 1
    $\begingroup$ 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 Sep 27 '18 at 13:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.