论文标题
2D自动保守动力学系统的可集成和可累积电位
Integrable and Superintegrable Potentials of 2d Autonomous Conservative Dynamical Systems
论文作者
论文摘要
我们考虑$ i = k_ {ab}（t，q）\ dot {q}^{a} \ dot {q}^{q}^{b}+k_ {a}（a a}（t，t，t，t，t，q）\ dot {q}^{q}^{a}+k（a a}+n $ d的$ d，后者导致偏微分方程的系统，该方程涉及张量$ k_ {ab}（t，q）$，$ k_ {a}（t，q）$，$ k（t，q）$以及动态方程的动态量。这些方程式分为两组。第一组仅涉及几何量的配置空间，第二组包含这些数量与动态场的相互作用。提出了一个定理，该定理根据配置空间中的动力学指标的插图提供了方程系统的系统解。该解决方案是几何和协变量，适用于更高的维度和弯曲空间。结果应用于二维（2D）自主保守的牛顿潜力的简单但有趣的情况。发现有两类的2D集成电位，并且在两个类别中都有可占地的电位。我们恢复了大多数以前的主要结果，这些结果已通过一种单一的和系统的方式通过各种方法获得。
We consider the generic quadratic first integral (QFI) of the form $I=K_{ab}(t,q)\dot{q}^{a}\dot{q}^{b}+K_{a}(t,q)\dot{q}^{a}+K(t,q)$ and require the condition $dI/dt=0$. The latter results in a system of partial differential equations which involve the tensors $K_{ab}(t,q)$, $K_{a}(t,q)$, $K(t,q)$ and the dynamical quantities of the dynamical equations. These equations divide in two sets. The first set involves only geometric quantities of the configuration space and the second set contains the interaction of these quantities with the dynamical fields. A theorem is presented which provides a systematic solution of the system of equations in terms of the collineations of the kinetic metric in the configuration space. This solution being geometric and covariant, applies to higher dimensions and curved spaces. The results are applied to the simple but interesting case of two-dimensional (2d) autonomous conservative Newtonian potentials. It is found that there are two classes of 2d integrable potentials and that superintegrable potentials exist in both classes. We recover most main previous results, which have been obtained by various methods, in a single and systematic way.
