AI 中文总结
本文针对COLCS流形建立哈密顿系统可积性框架，引入COLCS括号与李型可积性定理，研究标度对称性的性质，推广了LCS与余辛流形的相关理论。
AI 中文摘要
我们为**con-locally conformal symplectic（COLCS，共局部共形辛）流形**上的哈密顿动力学建立了可积性框架，COLCS流形是奇数维四元组$(M,\boldsymbol{\theta},\boldsymbol{\theta},\boldsymbol{\theta})$，其中$d\boldsymbol{\theta}=\boldsymbol{\theta}\boldsymbol{\theta}$，闭1-形式$\boldsymbol{\theta}$确定了一个余维1分布，$\boldsymbol{\theta}$在该分布上非退化，且Reeb向量场$R$满足$\boldsymbol{\theta}_R\boldsymbol{\theta}=\boldsymbol{\theta}_R\boldsymbol{\theta}=0$、$\boldsymbol{\theta}_R\boldsymbol{\theta}=1$。该类流形同时推广了LCS流形和余辛流形，为带扭曲微分$d^\theta=d-\theta\boldsymbol{\theta}$的时变哈密顿系统提供了自然舞台。我们在$C^\theta(M)$上引入COLCS括号，证明其为李括号，且在$\theta$-强函数（满足$\theta(X_H)=0$）和$R$-强函数（满足$R(H)=0$）的子代数上诱导泊松结构。随后建立了李型可积性定理：给定$2n-k$个函数独立的首次积分，其中$k$个为$\theta$-强函数，且在COLCS括号下生成可解李代数，则流在公共水平集上可通过二次积分实现可积。最后，我们研究了度为$(\boldsymbol{\theta},\boldsymbol{\theta},\boldsymbol{\theta})$的标度对称性，其定义为$\boldsymbol{\theta}_X\boldsymbol{\theta}=\boldsymbol{\theta}\boldsymbol{\theta}$、$\boldsymbol{\theta}_X H=\boldsymbol{\theta} H$、$\boldsymbol{\theta}_X\boldsymbol{\theta}=\boldsymbol{\theta}\boldsymbol{\theta}$：它们以$(\boldsymbol{\theta}-\boldsymbol{\theta})$标度$X_H$，生成首次积分族，并隐含$\boldsymbol{\theta}$和$\boldsymbol{\theta}$的结构基元。
英文摘要
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 $η$.
