Variational Approach and Deformed Derivatives

TL;DR

This paper extends the calculus of variations to include deformed derivatives in Lagrangian systems, leading to generalized Euler-Lagrange equations and Hamiltonian formalism compatible with nonlinear and mass-dependent dynamics.

Contribution

It introduces a novel variational framework incorporating deformed derivatives into classical and field Lagrangians, expanding the theoretical tools for complex systems.

Findings

Derived extended Euler-Lagrange equations with deformed derivatives.

Formulated a generalized Hamiltonian formalism for deformed systems.

Presented examples demonstrating applications and conserved currents.

Abstract

Recently, we have demonstrated that there exists a possible relationship between q-deformed algebras in two different contexts of Statistical Mechanics, namely, the Tsallis' framework and the Kaniadakis' scenario, with a local form of fractional-derivative operators for fractal media, the so-called Hausdorff derivatives, mapped into a continuous medium with a fractal measure. Here, in this paper, we present an extension of the traditional calculus of variations for systems containing deformed-derivatives embedded into the Lagrangian and the Lagrangian densities for classical and field systems. The results extend the classical Euler-Lagrange equations and the Hamiltonian formalism. The resulting dynamical equations seem to be compatible with those found in the literature, specially with mass-dependent and with nonlinear equations for systems in classical and quantum mechanics. Examples are presented to illustrate applications of the formulation. Also, the conserved Nether current, are worked out.