How is an optimized path of classical mechanics affected by random noise?

Abstract

The variational principle is one of important guiding principles in physics. Classical equations of motion of particle can be formulated so as to give the optimized path of an action. However, when there exist uncontrollable degrees of freedom such as noise, the optimized path is affected and the original classical equations of motion may not correspond to the optimized path. The stochastic variational method (SVM) is a framework to calculate the modified optimized path by the effect of noise. This method has been developed to show that the Schrödinger equation can be derived from the classical action which leads to Newton's equation of motion by taking into account the modification of the optimized path due to noise. In this work, we will extend this idea to the case of the continuum media and show that the Euler equation of the ideal fluid is converted to the Navier-Stokes equation or the Gross-Pitaevskii equation in SVM. © Published under licence by IOP Publishing Ltd.