辛流形上的欧拉-拉格朗日上同调群
The Euler-Lagrange Cohomology Groups on Symplectic Manifolds
摘要：
给出并研究了辛流形$({\cal M}^{2n},ω)$上的Euler-Lagrange上同调群$H^{2k-1}$，$1 \leqslant k \leqslant n$的定义和性质.对于$k = 1$和$k = n$，它们分别同构于相应的de Rham上同调群$H_{dR}^1({\cal M}^{2n})$和$H_{dR}^{2n-1}({\cal M}^{2n})$。其他的欧拉-拉格朗日上同调群一般都不同于$({\cal M}^{2n},ω)$上的de Rham上同调群或调和上同调群。从上同调的观点给出了$({\cal M}^{2n},ω)$上的一般保体积方程。在特殊情况下，这些方程成为Hamilton力学中的正则方程。因此，Hamilton力学通过上同调得到了推广。
英文摘要：
The definition and properties of the Euler-Lagrange cohomology groups $H^{2k-1}$, $1 \leqslant k \leqslant n$, on a symplectic manifold $({\cal M}^{2n},ω)$ are given and studied. For $k = 1$ and $k = n$, they are isomorphic to the corresponding de Rham cohomology groups $H_{dR}^1({\cal M}^{2n})$ and $H_{dR}^{2n-1}({\cal M}^{2n})$, respectively. The other Euler-Lagrange cohomology groups are different from either the de Rham cohomology groups or the harmonic cohomology groups on $({\cal M}^{2n},ω)$, in general. The general volume-preserving equations on $({\cal M}^{2n},ω)$ are also presented from cohomological point of view. In the special cases, these equations become the ordinary canonical equations in the Hamilton mechanics. Therefore, the Hamilton mechanics has been generalized via the cohomology.
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