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 $(M,Ω,θ,η)$ where $dΩ=θ\wedgeΩ$, a closed $1$-form $η$ determines a codimension-one distribution on which $Ω$ is non-degenerate, and $R$ is the Reeb vector field satisfying $ι_RΩ=ι_Rθ=0$, $ι_Rη=1$. This class simultaneously generalises LCS and cosymplectic manifolds and provides a natural arena for time-dependent Hamiltonian systems with twisted differential $d^θ=d-θ\wedge$. The COLCS bracket is introduced on $C^\infty(M)$ and shown to be a Lie bracket that induces Poisson structures on the subalgebras of $θ$-strong ($θ(X_H)=0$) 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 $θ$-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 $(Λ,β,γ)$, defined by $L_XΩ=βΩ$, $L_XH=ΛH$, $L_Xη=γη$, are studied: they rescale $X_H$ by $(Λ-β)$, generate families of first integrals, and imply structural primitives for $Ω$ and $η$.