Mass and total energy of moving bodies in a quantified expansion

In this paper, we present a new formulation of Lorentz transformations using a metric that quantifies the space expansion. As a consequence, we sort out that the limiting velocity of moving bodies is decreasing together with the space expansion. A new adjustment of relativistic laws is added to incorporate the non static nature of space-time. The conservation of the physical laws at each step of the quantified expansion allows the obtaining of new formalisms for the physical laws, in particular when an object starts moving under any force, its total energy, momentum and mass are directly affected by the expansion of the space. An example of inelastic collision is studied and several conclusions derived, specially the example of fission of atoms leads to clear correlation between liberated energy and universe expansion, it turns out that the liberated energy is increasing together with the universe expansion.Comment: 28 page

Fractal Topology Foundations

In this paper, we introduce the foundation of a fractal topological space constructed via a family of nested topological spaces endowed with subspace topologies, where the number of topological spaces involved in this family is related to the appearance of new structures on it. The greater the number of topological spaces we use, the stronger the subspace topologies we obtain. The fractal manifold model is brought up as an illustration of space that is locally homeomorphic to the fractal topological space.Comment: 20 page

Non-differentiable variational principles

We develop a calculus of variations for functionals which are defined on a set of non differentiable curves. We first extend the classical differential calculus in a quantum calculus, which allows us to define a complex operator, called the scale derivative, which is the non differentiable analogue of the classical derivative. We then define the notion of extremals for our functionals and obtain a characterization in term of a generalized Euler-Lagrange equation. We finally prove that solutions of the Schr\"odinger equation can be obtained as extremals of a non differentiable variational principle, leading to an extended Hamilton's principle of least action for quantum mechanics. We compare this approach with the scale relativity theory of Nottale, which assumes a fractal structure of space-time.Comment: 20 page

Fractional Vector Calculus and Fractional Maxwell's Equations

The theory of derivatives and integrals of non-integer order goes back to Leibniz, Liouville, Grunwald, Letnikov and Riemann. The history of fractional vector calculus (FVC) has only 10 years. The main approaches to formulate a FVC, which are used in the physics during the past few years, will be briefly described in this paper. We solve some problems of consistent formulations of FVC by using a fractional generalization of the Fundamental Theorem of Calculus. We define the differential and integral vector operations. The fractional Green's, Stokes' and Gauss's theorems are formulated. The proofs of these theorems are realized for simplest regions. A fractional generalization of exterior differential calculus of differential forms is discussed. Fractional nonlocal Maxwell's equations and the corresponding fractional wave equations are considered.Comment: 42 pages, LaTe

Scale calculus and the Schrodinger equation

We introduce the scale calculus, which generalizes the classical differential calculus to non differentiable functions. The new derivative is called the scale difference operator. We also introduce the notions of fractal functions, minimal resolution, and quantum representation of a non differentiable function. We then define a scale quantization procedure for classical Lagrangian systems inspired by the Scale relativity theory developped by Nottale. We prove that the scale quantization of Newtionian mechanics is a non linear Schrodinger equation. Under some specific assumptions, we obtain the classical linear Schrodinger equation.Comment: 49 page