Periodic first integrals for Hamiltonian systems of Lie type (Q2891005)

Let \((P, \omega )\) be a symplectic manifold and \(X\) a vector field on \(\mathbb{R}\times P\). Then \(X\) is called a \textit{Hamiltonian vector field of Lie type} if \(X(t, x)=\sum _{\alpha =1}^n b_{\alpha }(t)X_{\alpha }(x), x\in P\), with \((X_{\alpha })\) Hamiltonian vector fields on \(P\) spanning an \(n\)-dimensional real Lie algebra and \((b_{\alpha })\) smooth real functions. If \((b_{\alpha })\) are all periodic with the same period then \(X\) is called periodic. The present paper provides a general criterion for the existence of periodic first integrals for periodic Hamiltonian vector fields of Lie type. The Floquet theory is used to derive general conditions for the existence of a Poisson algebra of periodic first integrals. A very interesting application is considered for the Milne-Pinney system that describes the time evolution of an isotonic oscillator.