论文标题
自主保守动力学系统的二次二次积分
Quadratic first integrals of autonomous conservative dynamical systems
论文作者
论文摘要
自主动力系统由二阶微分方程的系统描述，该系统的解决方案提供了系统的轨迹。通过使用第一积分（FIS）来促进该解决方案，这些（FIS）用于减少微分方程系统的顺序，如果有足够的情况，则可以确定解决方案。因此，重要的是有一种系统的方法来确定FIS。另一方面，一个二阶微分方程的系统定义了动能，该动能提供了一个对称的二阶张量，称为系统的动力学指标。通过其对称性，该度量使场景众多的差异几何方法进入了场景，因此显然应该设法将FIS的确定与动力学指标的对称性联系起来。这项工作的主题是提供一个实现这种情况的定理。我们遵循的方法考虑了$ i = k_ {ab}（t，q^{c}）\ dot {q}^{a} \ dot {q}^{q}^{b}^{b} +k_} +k_ {a}（t，q^{c} {c} {c} {c} {其中$ k_ {ab}（t，q^{c}），k_ {a}（t，q^{c}），k（t，q^{c}）$是未知的张量量，需要$ di/dt = 0 $。这种条件导致涉及$ i $系数的微分方程系统，其解决方案提供了该形式的所有可能的二次FIS。我们演示了定理在测量方程的经典案例和广义开普勒电位中的应用。我们还获得并讨论了时间依赖的FI。
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.
