Forced Oscillations for Lagrangian and Hamiltonian Systems and the Nonhomogeneous Dirichlet Problem for Semilinear Elliptic Equations Involving Critical Exponents

拉格朗日和哈密顿系统的受迫振荡以及涉及临界指数的半线性椭圆方程的非齐次狄利克雷问题

• 批准号: 9003149
• 负责人: Gabriella Tarantello
• 金额: $ 4.23万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-06-01 至 1992-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9003149&HistoricalAwards=false
• 关键词: Forced; Oscillations; Lagrangian; Hamiltonian; Systems

Several problems in the field of differential equations will be under investigation during the term of this project. They concern systems of differential equations from the physical sciences, namely Lagrangian equations representing forced motion of a mechanical system and the more general Hamiltonian systems. A system of forced pendulums would give rise to the first class of equations, whereas the motion of bodies under gravitational forces often uses the Hamiltonian structure. The Lagrangian equations involved in this study have periodic coefficients and forcing functions. It is natural to inquire whether of not there will exist periodic solutions under such assumptions. As with many equations modeling phenomena from the physical world, the Lagrangians result from a principle of energy or action minimization. One seeks to find critical points of a certain integral with admissible functions taken from a Hilbert space. Depending on the growth of the potential, there may be many distinct periodic solutions, whereas evidence suggests that for periodic potentials, only finitely many periodic solutions may exist. A measure of the number of such solutions in terms the structure of the potential will be sought. Work on Hamiltonian systems will follow a similar vein concentrating on the case of periodic Hamiltonians. Solutions of these systems are known as subharmonics. There are known to be periodic subharmonics under very general conditions. The most important subharmonics are those with minimal periods. This research will look for a description of solutions with minimal periods in cases where additional symmetry assumptions are made on the Hamiltonian. Solutions of Hamiltonian systems must lie on surfaces of constant energy. A related goal of this work will be to determine which surfaces of the Hamiltonian carry a periodic orbit.

微分方程领域中的几个问题将 在本项目期间接受调查。 他们 从物理学角度考虑微分方程组 科学，即拉格朗日方程表示受迫运动 力学系统和更一般的哈密尔顿系统。 一个强制性的制度会产生第一个阶级， 而物体在引力作用下的运动 力通常使用哈密顿结构。 本研究中涉及的拉格朗日方程具有 周期系数和强迫函数。 是很自然 询问是否存在周期解 在这样的假设下。 与许多方程建模一样 从物理世界的现象，拉格朗日产生于一个 能量或作用最小化原理。 人们试图找到 一类容许函数积分的临界点 从希尔伯特空间中提取。 根据增长的 可能存在许多不同的周期解，而 有证据表明，对于周期电位， 可能存在许多周期解。 一种衡量 这样的解决方案在结构方面的潜力将是 寻找。 对哈密尔顿系统的研究也将遵循类似的思路 集中在周期哈密顿量的情况下。 解 这些系统被称为次谐波。 据了解， 在非常一般的条件下的周期次谐波。 最 重要的次谐波是具有最小周期的次谐波。 这 研究将寻找解决方案的描述， 周期的情况下，额外的对称性假设， 关于Hamilton 哈密顿系统的解必须依赖于 恒定能量的表面。 这项工作的一个相关目标将 是为了确定哪些表面的哈密尔顿携带一个 周期轨道