Maciejewski Andrzej J, Przybylska Maria
Janusz Gil Institute of Astronomy, University of Zielona Góra, Licealna 9, 65-417, Zielona Góra, Poland.
Institute of Physics, University of Zielona Góra, Licealna 9, 65-417, Zielona Góra, Poland
Philos Trans A Math Phys Eng Sci. 2018 Sep 17;376(2131):20170425. doi: 10.1098/rsta.2017.0425.
In this paper, we investigate systems of several point masses moving in constant curvature two-dimensional manifolds and subjected to certain holonomic constraints. We show that in certain cases these systems can be considered as rigid bodies in Euclidean and pseudo-Euclidean three-dimensional spaces with points which can move along a curve fixed in the body. We derive the equations of motion which are Hamiltonian with respect to a certain degenerated Poisson bracket. Moreover, we have found several integrable cases of described models. For one of them, we give the necessary and sufficient conditions for the integrability.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.
在本文中，我们研究了多个质点在常曲率二维流形中运动并受某些完整约束的系统。我们表明，在某些情况下，这些系统可被视为欧几里得和伪欧几里得三维空间中的刚体，其质点可沿刚体中固定的曲线移动。我们推导了相对于某个退化泊松括号为哈密顿形式的运动方程。此外，我们还找到了所描述模型的几个可积情形。对于其中一种情形，我们给出了可积性的充分必要条件。本文是主题为“有限维可积系统：新趋势与方法”的一部分。
