Spearheaded by the recent efforts to derive stochastic geophysical fluid dynamics models, we present a general framework for introducing stochasticity into variational principles through the concept of a semi-martingale driven variational principle and constraining the component variables to be compatible with the driving semi-martingale. Within this framework and the corresponding choice of constraints, the Euler–Poincaré equation can be easily deduced. We show that the deterministic theory is a special case of this class of stochastic variational principles. Moreover, this is a natural framework that enables us to correctly characterize the pressure term in incompressible stochastic fluid models. Other general constraints can also be incorporated as long as they are compatible with the driving semi-martingale.

在最近为推导随机地球物理流体动力学模型所做的努力的带动下，我们提出了一个一般框架，该框架通过半semi驱动的变分原理的概念并将随机变量约束为与驱动半semi驱动的相容性，从而将随机性引入变分原理。在此框架和相应的约束条件选择下，可以轻松推导出Euler-Poincaré方程。我们证明，确定性理论是这类随机变分原理的特例。而且，这是一个自然的框架，使我们能够正确地描述不可压缩的随机流体模型中的压力项。只要它们与半自动驾驶兼容，也可以并入其他一般约束。