Dynamical Systems

动力系统

• 批准号: 0074294
• 负责人: John Mather
• 金额: $ 35.1万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0074294&HistoricalAwards=false
• 关键词: Dynamical; Systems

This proposal has two parts. Mather's part concerns using variational methods to prove the existence of "wandering" orbits in Hamiltonian systems. The sort of thing he hopes to prove are illustrated by the conjectures which go under the name "Arnold diffusion". Forni's part concerns area preserving flows on surfaces. He intends to study two conjectures: an extension of the KAM theorem, where Kronecker flows are replaced by area preserving flows on surfaces of higher genus; and a conjecture that would imply that the convergence in Birkhoff's ergodic theorem takes place with polynomial-like decay. Both parts of this proposal concern generalized KAM theory, which has important applications to questions of stability and randomness in conservative physical systems. For example, the long-term dynamics of the solar system is far from being understood, and this proposal could contribute to mathematician's understanding.

这项建议有两个部分。 马瑟的部分涉及使用变分方法来证明存在的“徘徊”轨道的哈密顿系统。 他希望证明的那类事情可以通过名为“阿诺德扩散”的猜想来说明。 Forni的部分涉及表面上的面积保持流。 他打算研究两个apturtures：一个扩展的卡姆定理，其中克罗内克流被取代的面积保持流动的表面较高的属;和一个猜想，这将意味着收敛在伯克霍夫的遍历定理发生多项式样衰减。这两个部分的建议涉及广义KAM理论，它有重要的应用保守的物理系统的稳定性和随机性的问题。 例如，太阳系的长期动力学还远未被理解，这一提议可能有助于数学家的理解。