The Hamilton–Jacobi Analysis of Powers of Singular Lagrangians: A Connection Between the Modified Schrödinger and the Navier–Stokes Equations

Abstract

Non-standard Lagrangians have gained recently an increasing interest in the theory of nonlinear differential equations, classical and quantum nonlinear dynamical systems. In this work, we discuss a number of dynamical systems characterized by powers of singular Lagrangians identified to non-standard Lagrangians based on the resulting Hamilton–Jacobi equation. A number of dynamical problems were addressed and a number of statements which support the non-standard Lagrangians formalism were postulated. After connecting the action to a certain complex wave function and in particular for the case of linear potentials, a link is established between the resulting modified Schrodinger equation which describes specific classes of quantum mechanical systems and the Navier–Stokes equation which describes the motion of viscous fluid matters.