具有固定能量的约束自治保守动力系统的二次首次积分
Quadratic first integrals of constrained autonomous conservative dynamical systems with fixed energy
摘要：
考虑自治保守动力系统，其约束条件是系统的总能量具有特定值。我们证明了一个定理，它提供了二次第一积分（QFIs），时间依赖性和自主性，这些系统的对称性（共形Killing矢量和共形Killing张量）的动力学度量。证明了有三种类型的合格金融机构，并为每一种类型，我们给出了明确的公式计算。研究还表明，当考虑自治的合格金融机构时，我们恢复了以前工作的已知结果。对于零势函数，我们给出了约束测地线的情形，并得到了计算其QFI的公式。该定理适用于两种情况。在第一种情况下，我们确定的潜力，承认三种类型的合格金融机构的第二。我们恢复了Ermakov型的超可积势和一个新的可积势，其零能量和零QFI的轨迹是圆。在第二种情况下，我们集成了一族二维共形平坦度量的约束测地线方程。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
