Eulerian versus Lagrangian variational principles in stochastic mechanics

Ettore Aldrovandi, Daniela Dohrn, Francesco Guerra

Marra R, Maurizio Serva

Piotr Garbaczewski

R. Marra, Maurizio Serva

We propose an extension to the relativistic case of the stochastic variational principles both of Lagrangian and Eulerian type. The action we use is the mean classical action evaluated on the paths of relativistic covariant diffusions. The resulting equations of motion are the relativistic stochastic Lagrange equations Nous proposons une extension au cas relativiste des principes variationnels de type lagrangien et eulerien. Nous utilisons comme action l'action moyenne classique evaluee sur les trajectoires des diffusions covariantes relativistes. Les equations du mouvement obtenues sont les equations de Lagrange stochastiques relativistes

J.C. Zambrini

Jianhao M. Yang

Stochastic mechanics is regarded as a physical theory to explain quantum mechanics with classical terms such that some of the quantum mechanics paradoxes can be avoided. Here, we propose a new variational principle to uncover more insights on stochastic mechanics. According to this principle, information measures, such as relative entropy and Fisher information, are imposed as constraints on top of the least action principle. This principle not only recovers Nelson’s theory and, consequently, the Schrödinger equation but also clears an unresolved issue in stochastic mechanics on why multiple Lagrangians can be used in the variational method and yield the same theory. The concept of forward and backward paths provides an intuitive physical picture for stochastic mechanics. Each path configuration is considered as a degree of freedom and has its own law of dynamics. Thus, the variation principle proposed here can be a new tool to derive more advanced stochastic theory by including additional degrees of freedom in the theory. The structure of Lagrangian developed here shows that some terms in the Lagrangian are originated from information constraints. This suggests that a Lagrangian may need to include both physical and informational terms in order to have a complete description of the dynamics of a physical system.