Variational Approach and Deformed Derivatives

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.

13 pages. arXiv admin note: text overlap with arXiv:1502.07606

References in corpus (2)

• A possible deformed algebra and calculus inspired in nonextensive thermostatistics
• On a connection between a class of q-deformed algebras and the Hausdorff derivative in a medium with fractal metric

Cited by in corpus (7)

• On the nature of the conformable derivative and its applications to physics
• Dual Conformable Derivative: Definition, Simple Properties and Perspectives for Applications
• Coupled VAE: Improved Accuracy and Robustness of a Variational Autoencoder
• Time-fractional Caputo derivative versus other integro-differential operators in generalized Fokker-Planck and generalized Langevin equations
• Structural scale $q-$derivative and the LLG-Equation in a scenario with fractionality
• A deformed derivative model for turbulent diffusion of contaminants in the atmosphere
• Existence of positive solutions for a class of conformable fractional differential equations with integral boundary conditions and a parameter