8 research outputs found
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
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
The Relativistic Electrodynamics Least Action Principles Revisited: New Charged Point Particle and Hadronic String Models Analysis
The classical relativistic least action principle is revisited from the
vacuum field theory approach. New physically motivated versions of relativistic
Lorentz type forces are derived, a new relativistic hadronic string model is
proposed and analyzed in detail.Comment: n/
The Vacuum Structure, Special Relativity and Quantum Mechanics Revisited: a Field Theory No-Geometry Approach within the Lagrangian and Hamiltonian Formalisms. Part 2
The main fundamental principles characterizing the vacuum field structure are
formulated and the modeling of the related vacuum medium and charged point
particle dynamics by means of devised field theoretic tools are analyzed. The
work is devoted to studying the vacuum structure, special relativity,
electrodynamics of interacting charged point particles and quantum mechanics,
and is a continuation of \cite{BPT,BRT1}. Based on the vacuum field theory
no-geometry approach, the Lagrangian and Hamiltonian reformulation of some
alternative classical electrodynamics models is devised. The Dirac type
quantization procedure, based on the canonical Hamiltonian formulation, is
developed for some alternative electrodynamics models. Within an approach
developed a possibility of the combined description both of electrodynamics and
gravity is analyzed.Comment: 11 page