Integrability of Hamiltonian systems on COLCS Manifold

Abstract

We develop an integrability framework for Hamiltonian dynamics on \emph{con-locally conformal symplectic} (COLCS) manifolds, odd-dimensional quadruples $(M,\Omega ,\theta ,\eta )$ where $d\Omega =\theta \wedge \Omega$, a closed $1$-form $\eta$ determines a codimension-one distribution on which $\Omega$ is non-degenerate, and $R$ is the Reeb vector field satisfying ${\iota }_R\Omega ={\iota }_R\theta =0$, ${\iota }_R\eta =1$. This class simultaneously generalises LCS and cosymplectic manifolds and provides a natural arena for time-dependent Hamiltonian systems with twisted differential $d^{\theta }=d-\theta \wedge$. The COLCS bracket is introduced on $C^{\infty }(M)$ and shown to be a Lie bracket that induces Poisson structures on the subalgebras of $\theta$-strong ($\theta (X_H)=0$) and $R$-strong ($R(H)=0$) functions. A Lie-type integrability theorem is then established: given $2n-k$ functionally independent first integrals with $k$ of them $\theta$-strong and generating a solvable Lie algebra under the COLCS bracket, the flow is integrable by quadratures on the common level set. Finally, scaling symmetries of degree $(\Lambda ,\beta ,\gamma )$, defined by $L_X\Omega =\beta \Omega$, $L_{X}H=\Lambda H$, $L_X\eta =\gamma \eta$, are studied: they rescale $X_H$ by $(\Lambda -\beta )$, generate families of first integrals, and imply structural primitives for $\Omega$ and $\eta$.