We develop the non-di erentiable embedding theory of di erential operators and Lagrangian systems using a new operator on non-di erentiable functions. We then construct the corresponding calculus of variations and we derive the associated non-di erentiable Euler-Lagrange equation, and apply this formalism to the study of PDE's. First, we extend the characteristics method to the non-di erentiabl...

We perform a systematic multiscale analysis for the 2-D incompressible Euler equation with rapidly oscillating initial data using a Lagrangian approach. The Lagrangian formulation enables us to capture the propagation of the multiscale solution in a natural way. By making an appropriate multiscale expansion in the vorticity-stream function formulation, we derive a well-posed homogenized equatio...

An approximate method for describing the evolution of solitonlike initial conditions to solitons for the sine-Gordon equation is developed. This method is based on using a solitonlike pulse with variable parameters in an averaged Lagrangian for the sine-Gordon equation. This averaged Lagrangian is then used to determine ordinary differential equations governing the evolution of the pulse parame...

The Principle of Least Action is used with a simple Lagrangian density, involving second-order derivatives of the wave function, to obtain the Schrödinger equation. A Hamiltonian density obtained from this simple Lagrangian density shows that Hamilton’s equations also give the Schrödinger equation. This simple Lagrangian density is equivalent to a standard Lagrangian density with first-order de...

— This paper is a contribution to the general program of embedding theories of dynamical systems. Following our previous work on the Stochastic embedding theory developed with S. Darses, we define the fractional embedding of differential operators and ordinary differential equations. We construct an operator combining in a symmetric way the left and right (Riemann-Liouville) fractional derivati...

— This paper is a contribution to the general program of embedding theories of dynamical systems. Following our previous work on the Stochastic embedding theory developed with S. Darses, we define the fractional embedding of differential operators and ordinary differential equations. We construct an operator combining in a symmetric way the left and right (Riemann-Liouville) fractional derivati...

II Symmetries in Classical Particle Mechanics 4 1 Hamiltonian and Lagrangian Formulations of Classical Particle Mechanics 5 1.1 Basic De nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.1 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.2 Virtual Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.3 Generalized...

A method is proposed of constructing quantum correlators for a general gauge system whose classical equations of motion do not necessarily follow from the least action principle. The idea of the method is in assigning a certain BRST operator Ω̂ to any classical equations of motion, Lagrangian or not. The generating functional of Green’s functions is defined by the equation Ω̂Z(J) = 0 that is redu...