Indices and Stability of the Lagrangian System on Riemannian Manifold
Indices and Stability of the Lagrangian System on Riemannian Manifold
复制标题
黎曼流形上拉格朗日系统的指标和稳定性
DOI：
10.1007/s10114-020-9311-7
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
Zhu Gaosheng
中科院分区：
文献类型：
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作者：
Zhu Gaosheng
设m>；=1为整数，M为m维紧致黎曼流形。首先明确地给出了拉格朗日系统在临界点x…d/dt L-q(t，x，x)-L-p(t，x，x)-p(t，x，x)=0的线性化Poincare映射，然后利用不诉诸酉平凡化的平行传输方法证明了X的Morse指标和Maslov型指标是很好地定义的，并建立了Morse指标和Maslov型指标之间的关系，最后得到了M的临界点和方向不稳定的判据，并得到了这两个Maslov型指标的计算公式.
In this paper, let m >= 1 be an integer, M be an m-dimensional compact Riemannian manifold. Firstly the linearized Poincare map of the Lagrangian system at critical point x..d/dt L-q(t, x, x) - L-p(t, x, x) = 0..is explicitly given, then we prove that Morse index and Maslov-type index of x are well defined whether the manifold M is orientable or not via the parallel transport method which makes no appeal to unitary trivialization and establish the relation of Morse index and Maslov-type index, finally derive a criterion for the instability of critical point and orientation of M and obtain the formula for two Maslov-type indices.