We derive and study stochastic dissipative dynamics on coadjoint orbits by incorporating noise and dissipation into mechanical systems arising from the theory of reduction by symmetry, including a semidirect product extension. Random attractors are found for this general class of systems when the Lie algebra is semi-simple, provided the top Lyapunov exponent is positive. We study in details two canonical examples, the free rigid body and the heavy top, whose stochastic integrable reductions are found and numerical simulations of their random attractors are shown.

中文翻译：

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共生轨道上的噪声和耗散。

我们通过将噪声和耗散并入由对称性减少理论（包括半直接乘积扩展）引起的机械系统中，得出并研究共伴轨道上的随机耗散动力学。只要李雅普诺夫指数最高为正，当李代数为半简单时，就可以为这类一般系统找到随机吸引子。我们详细研究了两个典型的例子，自由刚体和重顶，发现了它们的随机可积约简，并给出了它们的随机吸引子的数值模拟。