Convexity Methods In Hamiltonian Mechanics
Convexity Methods In Hamiltonian Mechanics
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DOI：
10.1007/978-3-642-74331-3
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发表时间：
1990
期刊：
影响因子：
--
通讯作者：
I. Ekeland
中科院分区：
文献类型：
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作者：
I. Ekeland
对于完全可积系统，周期解是通过检验得到的。对于不可积系统，例如天体力学中的三体问题，它们是通过摄动理论找到的：问题中有一个小参数€，例如摄动体的质量，当€= 0时，系统变得完全可积。然后试图证明它的周期解在€# 0足够小的情况下仍然存在。庞加莱还引入了全局方法，依靠流的拓扑性质，以及它保持2-form L~= L dPi 1\dqi’的事实。他在这个方向上获得的最著名的结果是他的最后一个几何定理，该定理指出，使内圆和外圆以相反方向旋转的环空的保面积映射必须有两个不动点。现在又出现了另一个古老的主题：最小作用原理。说明了哈密顿系统的周期解是闭曲线上合适积分的极值。换句话说，这个问题是变分的。费马知道这个事实，莫珀蒂把它写进了汉密尔顿的形式主义。尽管最小作用原理具有巨大的美学吸引力，但它在哈密顿力学中几乎没有什么影响。当然，有一个例外，那就是埃米·诺特定理，它将运动的积分与方程的对称性联系起来。但直到最近，还没有人用变分方法找到周期解。
In the case of completely integrable systems, periodic solutions are found by inspection. For nonintegrable systems, such as the three-body problem in celestial mechanics, they are found by perturbation theory: there is a small parameter€ in the problem, the mass of the perturbing body for instance, and for€= 0 the system becomes completely integrable. One then tries to show that its periodic solutions will subsist for€-# 0 small enough. Poincare also introduced global methods, relying on the topological properties of the flow, and the fact that it preserves the 2-form L~= l dPi 1\dqi'The most celebrated result he obtained in this direction is his last geometric theorem, which states that an area-preserving map of the annulus which rotates the inner circle and the outer circle in opposite directions must have two fixed points. And now another ancient theme appear: the least action principle. It states that the periodic solutions of a Hamiltonian system are extremals of a suitable integral over closed curves. In other words, the problem is variational. This fact was known to Fermat, and Maupertuis put it in the Hamiltonian formalism. In spite of its great aesthetic appeal, the least action principle has had little impact in Hamiltonian mechanics. There is, of course, one exception, Emmy Noether's theorem, which relates integrals ofthe motion to symmetries of the equations. But until recently, no periodic solution had ever been found by variational methods.