In order to apply the Marsden–Weinstein reduction, the action of the group $G$ must be free and proper. On the other hand, if I correctly understand, the M-W reduction obtained from a given group $G$ can be used to decrease the number of degrees of freedom of a Hamiltonian $H$, provided that the Hamiltonian flow of $H$ commutes with the action of $G$.


Could you please give an example of such a Hamiltonian $H$ and of such a group $G$, whose action is not proper?


Please, try to give an example in which $G$ has the lowest possible dimension: I mean, if it is possible, provide a 1-dimensional Lie group $G$.



Marsden–Weinstein: example of not proper action
(Comment $\to$ answer as requested.)


Let $G=\mathbf R$ act on the 2-torus $Z=\mathrm U(1)\times\mathrm U(1)$ by $g(z\_1,z\_2)=(e^{ig}z\_1, e^{i\pi g}z\_2)$. Lift the action to $T^\*Z$ and use any $G$-invariant $H$.


Explicitly $T^\*Z=\mathbf R^2\times Z\ni(p\_1,p\_2,z\_1,z\_2)$ where $G$ acts by the flow of $K=p\_1+\pi p\_2$ (not proper by Bourbaki, Topologie générale, Chap. III, §4, [Exercice 5](//books.google.com/books?id=12yuCBbwijUC&ots=YyfQxnCqiI&pg=PA74&dq=ferm%C3%A9)), and say $H=$ any function of $(p\_1,p\_2)$.


