论文标题
具有固定能量的受约束自主保守动力学系统的二次二次积分
Quadratic first integrals of constrained autonomous conservative dynamical systems with fixed energy
论文作者
论文摘要
我们考虑自主保守的动力系统，这些系统受到系统的总能量具有指定值的约束。我们证明了一个定理，它提供了这些系统的二次第一积分（QFI），这些定理是根据动力学指标的对称性（共形杀死向量和形式的杀伤向量和共形杀伤量）的角度。事实证明，QFI有三种类型，对于每种类型，我们为其计算提供了明确的公式。还表明，当考虑自主QFI时，我们会恢复以前工作的已知结果。对于零电位函数，我们具有受约束的测量学的情况，并获得了计算其QFI的公式。定理在两种情况下应用。在第一种情况下，我们确定允许三种QFI中第二种的电势。我们恢复了Ermakov类型的可更稳定势和一个新的集成电位，其零能量和零QFI的轨迹是圆圈。在第二种情况下，我们将约束的地质方程集成到二维形式平面度量的家族中。
We consider autonomous conservative dynamical systems which are constrained with the condition that the total energy of the system has a specified value. We prove a theorem which provides the quadratic first integrals (QFIs), time-dependent and autonomous, of these systems in terms of the symmetries (conformal Killing vectors and conformal Killing tensors) of the kinetic metric. It is proved that there are three types of QFIs and for each type we give explicit formulae for their computation. It is also shown that when the autonomous QFIs are considered, then we recover the known results of previous works. For zero potential function, we have the case of constrained geodesics and obtain formulae to compute their QFIs. The theorem is applied in two cases. In the first case, we determine potentials which admit the second of the three types of QFIs. We recover a superintegrable potential of the Ermakov type and a new integrable potential whose trajectories for zero energy and zero QFI are circles. In the second case, we integrate the constrained geodesic equations for a family of two-dimensional conformally flat metrics.
