论文标题
具有随机电势的天然哈密顿系统的非整合性和混乱
Non-integrability and chaos for natural Hamiltonian systems with a random potential
论文作者
论文摘要
考虑高斯随机潜力的合奏$ \ {v^l（q）\} _ {l = 1}^\ infty $在$ d $ d $维圆环上，基本上，$ v^l（q）$是真实价值的三角三角三角元的$ l $具有独立的标准常规变量。我们的主要结果确保了概率趋于$ l \ to \ infty $的概率，该动力系统与自然的汉密尔顿功能相关，由自然的汉密尔顿功能与随机电位定义，$ h^l：= \ frac12 | p |^2+ v^l（q）$，表现出许多与阳性vol-volume set neveriant tori的杂种区域。特别是，这些系统通常既无法与非脱位的第一积分也不可以集成。在任意紧凑的riemannian歧管的cotangent束上定义的随机天然汉密尔顿系统的类似结果。
Consider the ensemble of Gaussian random potentials $\{V^L(q)\}_{L=1}^\infty$ on the $d$-dimensional torus where, essentially, $V^L(q)$ is a real-valued trigonometric polynomial of degree $L$ whose coefficients are independent standard normal variables. Our main result ensures that, with a probability tending to 1 as $L\to\infty$, the dynamical system associated with the natural Hamiltonian function defined by this random potential, $H^L:=\frac12|p|^2+ V^L(q)$, exhibits a number of chaotic regions which coexist with a positive-volume set of invariant tori. In particular, these systems are typically neither integrable with non-degenerate first integrals nor ergodic. An analogous result for random natural Hamiltonian systems defined on the cotangent bundle of an arbitrary compact Riemannian manifold is presented too.
