自治保守动力系统的二次首次积分
Quadratic first integrals of autonomous conservative dynamical systems
摘要：
一个自治的动力系统是由一个二阶微分方程的系统，其解决方案给出了系统的轨迹。通过使用第一积分（FI）来促进解决方案，第一积分用于降低微分方程系统的阶数，并且如果有足够的积分，则用于确定解决方案。因此，重要的是存在一个系统的方法来确定金融机构。另一方面，二阶微分方程系统定义了一个动能，它提供了一个对称的二阶张量，称为系统的动力学度量。这个度规通过它的对称性将微分几何的许多方法带入场景，因此很明显，人们应该设法将FI的确定与动力学度规的对称性联系起来。这项工作的主题是提供一个定理，实现这种情况。我们遵循的方法考虑了$I=K_{ab}(t,q^{c})\dot{q}^{a}\dot{q}^{b}+K_{a}(t,q^{c})\dot{q}^{a} +K(t,q^{c})$形式的一般二次FI，其中$K_{ab}(t,q^{c}), K_{a}(t,q^{c}), K(t,q^{c})$是未知张量，需要$dI/dt = 0$。该条件导致涉及$I$的系数的微分方程系统，其解提供了该形式的所有可能的二次FI。我们证明了定理的应用中的经典情况下的测地线方程和广义开普勒势。我们还获得并讨论了时间依赖的FI。
英文摘要：
An autonomous dynamical system is described by a system of second order differential equations whose solution gives the trajectories of the system. The solution is facilitated by the use of first integrals (FIs) that are used to reduce the order of the system of differential equations and, if there are enough of them, to determine the solution. Therefore, it is important that there exists a systematic method to determine the FIs. On the other hand, a system of second order differential equations defines a kinetic energy, which provides a symmetric second order tensor called kinetic metric of the system. This metric via its symmetries brings into the scene the numerous methods of differential geometry and hence it is apparent that one should manage to relate the determination of the FIs to the symmetries of the kinetic metric. The subject of this work is to provide a theorem that realizes this scenario. The method we follow considers the generic quadratic FI of the form $I=K_{ab}(t,q^{c})\dot{q}^{a}\dot{q}^{b}+K_{a}(t,q^{c})\dot{q}^{a} +K(t,q^{c})$ where $K_{ab}(t,q^{c}), K_{a}(t,q^{c}), K(t,q^{c})$ are unknown tensor quantities and requires $dI/dt = 0$. This condition leads to a system of differential equations involving the coefficients of $I$ whose solution provides all possible quadratic FIs of this form. We demonstrate the application of the theorem in the classical cases of the geodesic equations and the generalized Kepler potential. We also obtain and discuss the time-dependent FIs.
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