Non-differentiable embedding of Lagrangian systems and partial differential equations

We develop the non-differentiable embedding theory of differential operators and Lagrangian systems using a new operator on non-differentiable functions. We then construct the corresponding calculus of variations and we derive the associated non-differentiable Euler–Lagrange equation, and apply this formalism to the study of PDEs. First, we extend the characteristics method to the non-differentiable case. We prove that non-differentiable characteristics for the Navier–Stokes equation correspond to extremals of an explicit non-differentiable Lagrangian system. Second, we prove that the solutions of the Schrödinger equation are non-differentiable extremals of the Newtonʼs Lagrangian.