Kirillov structures and reduction of Hamiltonian systems by scaling and standard symmetries (Q6590505)

Let \((S,\omega)\) be a symplectic manifold and \(H:S\to \mathbb{R}\) a Hamiltonian function. Assume that there is a Lie group \(G\) acting freely and symplectically on \(S\), so that \(H\) is \(G\)-invariant while \(S/G\) is a smooth manifold. Then, it is well-known that \(S/G\) inherits a Poisson structure from \((S,\omega)\) and that \(H\) descends to the quotient. \N\NThe present work is concerned with the existence of an additional \textit{scaling symmetry}, i.e., an action \(\phi: \mathbb{R}^\times \times S \to S\) such that \(\phi^\ast_s \omega=s.\omega\) and \(\phi^\ast_s H=s.H\) for any \(s\in \mathbb{R}^\times\). When the \(G\)- and \(\mathbb{R}^\times\)-actions commute, and the quotient \((S/G)/\mathbb{R}^\times \simeq (S/\mathbb{R}^\times)/G\) is a manifold, it can be naturally equipped with a \textit{Kirillov structure}. It is given by a real line bundle endowed with a Lie bracket (satisfying extra assumptions, cf. Definition 1 based on [\textit{A. A. Kirillov}, Russ. Math. Surv. 31, 55--75 (1976; Zbl 0357.58003)]) induced by the symplectic structure \(\omega\) on \(S\).\N\NThe Kirillov structures obtained either by reducing with respect to \(G\) then the scaling \(\mathbb{R}^\times\)-action, or first by reducing with the scaling symmetry and then by \(G\), are compared, cf. Section 6, and shown to be equivalent. The problem of lifting the flow of the (reduced) Kirillov system to a flow of the original symplectic Hamiltonian system \((S,\omega,H)\) is considered in Section 7. Two examples of Kirillov structures with an additional Hamiltonian are used to illustrate the theory: the two-dimensional harmonic oscillator and the projective cotangent bundle with a linear Hamiltonian.