# Research

#### Comment on «Dynamics with Infinitely Many Derivatives: The Initial Value Problem»[🔗]

Journal: Non-published

Date: June 8, 2021

Authors: Carlos Heredia & Josep Llosa

The definition of fractional operators usually resorts to Laplace transforms and convolution. Since the “natural” arena for the latter seems to be the space of generalised functions with support on a half-line, we pinpoint some useful notions extracted from Equations of Mathematical Physics – V. S. Vladimirov (sections from 10.2 to 10.4). Using these notions, we make some comments about the article «Dynamics with Infinitely Many Derivatives: The Initial Value Problem».

#### Non-local Lagrangian Mechanics: Noether theorem and Hamiltonian formalism

Journal: arXiv [hep-th] – 2105.10442

Date: May 24, 2021

Authors: Carlos Heredia & Josep Llosa

We study Lagrangian systems with a finite number of degrees of freedom that are non-local in time. We obtain an extension of Noether theorem and Noether identities to this kind of Lagrangians. A Hamiltonian formalism is then set up for this systems. n-order local Lagrangians can be treated as a particular case and the standard results for them are recovered. The method is then applied to several other cases, namely two examples of non-local oscillators and the p-adic particle.

#### Energy-momentum tensor for the electromagnetic field in a dispersive medium

Date: May 10, 2021

Authors: Carlos Heredia & Josep Llosa

On the basis of a non-local Lagrangian for Maxwell equations in a dispersive medium, the energy-momentum tensor of the field is derived. We obtain the Field equations through variational methods and an extension of Noether theorem for a non-local Lagrangian is obtained as well. The electromagnetic energy-momentum tensor obtained in the general context is then specialized to the case of a field with slowly varying amplitude on a rapidly oscillating carrier.

#### Poster session[🔗]: Energy-momentum tensor for the electromagnetic field in a dispersive medium

Date: Mar 5, 2020

Speaker: Carlos Heredia

This work aims to obtain the Electromagnetic Stress-Energy Tensor in a medium for a non-local theory. To get it, we generalize Minkowski electrodynamics to dispersive media. As a consequence of this generalization, the Lagrangian density becomes non-local due to the non-local dependencies of the magnetic permeability and electric permittivity. […] Then, we derive the field equations and, applying Noether’s theorem, the conserved energy-momentum tensor. Because non-local Lagrangians are seldom found in textbooks, we devote a non-local formalism to outline the field equations’ derivation and Noether’s theorem. […] To conclude, we study the obtained Belinfante Stress-Energy Tensor for plane wave solutions for a dispersive medium.

#### Hamilton-Jacobi formalism for non-involutive systems [🔗]

Journal: Non-published

Date: Jun 20, 2017

Authors: Carlos Heredia & Josep Mª Pons

This work aims to show our vision about the inconsistency of the integrability condition for non-involutive systems in the Hamilton-Jacobi formalism. A dimensional reduction is made to solve this issue, indicating that the reduced system contains the same dynamics as the original one. Moreover, to give consistency to our work, we will display this method in four examples.

#### Hamilton-Jacobi formalism for gauge theories [🔗]

Journal: Non-published

Date: Jan 30, 2016

Authors: Carlos Heredia & Josep Mª Pons

This work aims to study the Hamilton-Jacobi formalism for singular systems (or systems with gauge symmetries). We review the problem with these systems to understand the motivation to construct what is known as Dirac Hamiltonian or total Hamiltonian for the dynamics of constrained systems. We conclude with the study of the free particle in the Minkowski spacetime as an example.