# Non-Hamiltonian actions in physics

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 :)

• What exactly do you understand as a "Hamiltonian symplectic action"? A symplectic Lie group action which has a momentum map? – Tobias Diez Sep 27 '18 at 13:44
• 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). – Igor Khavkine Sep 27 '18 at 13:56

## 1 Answer

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).

• 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 – R Mary Sep 27 '18 at 13:12