arXiv subjects

Matheus J. Lazo

Publications and source records attributed to Matheus J. Lazo.

17 recordsLinked to original sources

arXiv2025

Tilt in quadratic gravity II

We investigate a tilted fluid component on a Bianchi V geometry in the theories of General Relativity (GR) and Quadratic Gravity (QG). The main objective of this work is the study of how the properties of matter can modify the future evolution of the attractors and their consequences on the regions of initial conditions of the solutions. As is well known, QG contains the Ruzmaikina-Ruzmaikin (RR) solution. This solution describes the slow-roll regime of Starobinsky's inflationary model, which is currently the best one due to the excellent agreement with Cosmic Microwave Background Radiation (CMBR) data. In QG, we found universes that can be attracted to the RR solution or recollapse toward the isotropic singularity attractor. If the Equation of State (EoS) parameter is ultra-radiative w>1/3, the tilt variable increases both in RR and Milne for QG or GR, respectively. In both cases, the fluid expansion and acceleration diverge, while the vorticity initially increases and then decreases to zero.

gr-qc↗

arXiv2024

Tilt in quadratic gravity

In this work, tilted source solutions in both Einstein-Hilbert General Relativity (GR) and Quadratic Gravity (QG) for the anisotropic Bianchi V model are addressed. Since the excellent CMBR match of Starobinsky's inflation with Planck's team measurements data, QG has acquired a prominent status in the effective sense, for sufficiently strong gravity fields. The main interest is in the numeric time evolution to the past towards the singularity and the behavior of the kinematic variables, vorticity, acceleration, and the expansion of this source substance. In QG we found that for universes with higher and smaller matter densities fall into the Kasner or isotropic singularity attractors to the past, respectively. We also found that the Kasner singularity attractor to the past has always zero vorticity, for both GR and QG theories. While for QG the isotropic singularity attractor may have divergent vorticity. For the set of assumptions and conditions supposed in this work, the isotropic singularity attractor, favors QG as compared to GR. Only in QG we were able to find a geometric singularity with divergences in all of the kinematic variables of the substance, decreasing to finite values to the future, upon time reversing. That is, we obtained an initial kinematic singularity substance, that approaches a perfect fluid source.

gr-qc↗

arXiv2021

Método de Monte Carlo aplicado ao Cálculo Fracionário

This article analyzes and develops a method to solve fractional ordinary differential equations using the Monte Carlo Method. A numerical simulation is performed for some differential equations, comparing the results with what exists in the mathematical literature. The Python language is used to create computational models.

math.NA↗

arXiv2021

Calculus of variations and optimal control for generalized functions

We present an extension of some results of higher order calculus of variations and optimal control to generalized functions. The framework is the category of generalized smooth functions, which includes Schwartz distributions, while sharing many nonlinear properties with ordinary smooth functions. We prove the higher order Euler-Lagrange equations, the D'Alembert principle in differential form, the du Bois-Reymond optimality condition and the Noether's theorem. We start the theory of optimal control proving a weak form of the Pontryagin maximum principle and the Noether's theorem for optimal control. We close with a study of a singularly variable length pendulum, oscillations damped by two media and the Pais-Uhlenbeck oscillator with singular frequencies.

math.FA↗

arXiv2021

Generalized nonconservative gravitational field equations from Herglotz action principle

We present an alternative nonconservative gravitational theory based on the Herglotz variational principle in a fully covariant form. The present model may be seen as an improvement of the theory proposed in Ref. [Lazo et al, Phys. Rev. D 95, 101501 (2017)], whose resulting theory is meaningful just in particular coordinate systems. In the present work, we report a new theory that is free from such a restriction. It is also obtained using the Herglotz variational principle and by taking advantage of the restricted equivalence between Lagrangian functions in the scope of such action principle. The more restricted class of equivalent Lagrangian functions, in comparison with the Hamilton variational principle, is the key point to find a Lagrangian that furnishes a new alternative gravitational theory that is fully covariant. Once the equations that govern the dynamics of the gravitational field are obtained, a few simple cosmological models are investigated. It is found that the Herglotz gravitational field reduces to a single function that, under certain conditions, plays the role of the cosmological constant in general relativity, turning unnecessary the use of dark energy to explain the accelerated expansion of the universe. The linearized version of the theory is also investigated and it is verified that the theory shows a dissipative character in regard to gravitational waves. From observational data, in both scenarios, the Herglotz vector field is estimated.

gr-qc↗

arXiv2018

Dark energy with zero pressure: Accelerated expansion and large scale structure in action-dependent Lagrangian theories

We develop a cosmological model based on action-dependent Lagrangian theories. The main feature here is the nonconservation of the energy momentum tensor due to the nontrivial geometrical construction of the theory. We provide the basic set of equations necessary to study both the cosmological background expansion as well as the linear matter perturbation growth. We show that the simplest realization of the Universe as described by only one component is not viable as expected from the existing correspondence between this model and the case of viscous cosmological fluids. However, modeling the energy content of the Universe as composed by two pressureless fluids, i.e., one a typical cold dark matter fluid and the other a pressureless dark energy fluid which is responsible for driving the late-time acceleration expansion, is qualitatively compatible with observational data.

gr-qc↗

arXiv2018

An Action Principle for Action-dependent Lagrangians: toward an Action Principle to non-conservative systems

In this work, we propose an Action Principle for Action-dependent Lagrangian functions by generalizing the Herglotz variational problem to the case with several independent variables. We obtain a necessary condition for the extremum equivalent to the Euler-Lagrange equation and, through some examples, we show that this generalized Action Principle enables us to construct simple and physically meaningful Action-dependent Lagrangian functions for a wide range of non-conservative classical and quantum systems. Furthermore, when the dependence on the Action is removed, the traditional Action Principle for conservative systems is recovered.

math-ph↗

arXiv2017

Cosmology from a new non-conservative gravity

In this paper we present a cosmological model arising from a non-conservative gravitational theory proposed in [PRD 95, 101501(R) (2017)]. The novel feature where comparing with previous implementations of dissipative effects in gravity is the possible arising of such phenomena from a least action principle, so they are of a purely geometric nature. We derive the dynamical equations describing the behaviour of the cosmic background, considering a single fluid model composed by pressureles matter, whereas the dark energy is conceived as an outcome of the "geometric" dissipative process emerging in the model. Besides, adopting the synchronous gauge we obtain the first-order perturbative equations which shall describe the evolution of the matter perturbations within the linear regime.

gr-qc↗

arXiv2017

From an Action Principle for Action-dependent Lagrangians toward non-conservative Gravity: accelerating Universe without dark energy

In the present work, we propose an Action Principle for Action-dependent Lagrangians by generalizing the Herglotz variational problem for several independent variables. This Action Principle enables us to formulate Lagrangian densities for non-conservative fields. In special, from a Lagrangian depending linearly on the Action, we obtain a generalized Einstein's field equations for a non-conservative gravity and analyze some consequences of their solutions to cosmology and gravitational waves. We show that the non-conservative part of the field equations depends on a constant cosmological four-vector. Depending on this four-vector, the theory displays damped/amplified gravitational waves and an accelerating Universe without dark energy.

gr-qc↗

arXiv2016

Variational Calculus with Conformable Fractional Derivatives

Invariant conditions for conformable fractional problems of the calculus of variations under the presence of external forces in the dynamics are studied. Depending on the type of transformations considered, different necessary conditions of invariance are obtained. As particular cases, we prove fractional versions of Noether's symmetry theorem. Invariant conditions for fractional optimal control problems, using the Hamiltonian formalism, are also investigated. As an example of potential application in Physics, we show that with conformable derivatives it is possible to formulate an Action Principle for particles under frictional forces that is far simpler than the one obtained with classical fractional derivatives.

math.OC↗

arXiv2016

Fractional Noether's Theorem with Classical and Caputo Derivatives: constants of motion for non-conservative systems

Since the seminal work of Emmy Noether it is well know that all conservations laws in physics, \textrm{e.g.}, conservation of energy or conservation of momentum, are directly related to the invariance of the action under a family of transformations. However, the classical Noether's theorem can not yields informations about constants of motion for non-conservative systems since it is not possible to formulate physically meaningful Lagragians for this kind of systems in classical calculus of variation. On the other hand, in recent years the fractional calculus of variation within Lagrangians depending on fractional derivatives has emerged as an elegant alternative to study non-conservative systems. In the present work, we obtained a generalization of the Noether's theorem for Lagrangians depending on mixed classical and Caputo derivatives that can be used to obtain constants of motion for dissipative systems. In addition, we also obtained Noether's conditions for the fractional optimal control problem.

math.OC↗

arXiv2014

The action principle for dissipative systems

In the present work we redefine and generalize the action principle for dissipative systems proposed by Riewe by fixing the mathematical inconsistencies present in the original approach. In order to formulate a quadratic Lagrangian for non-conservative systems, the Lagrangian functions proposed depend on mixed integer order and fractional order derivatives. As examples, we formulate a quadratic Lagrangian for a particle under a frictional force proportional to the velocity, and to the classical problem of an accelerated point charge.

math-ph↗

arXiv2013

The Legendre Condition of the Fractional Calculus of Variations

Fractional operators play an important role in modeling nonlocal phenomena and problems involving coarse-grained and fractal spaces. The fractional calculus of variations with functionals depending on derivatives and/or integrals of noninteger order is a rather recent subject that is currently in fast development due to its applications in physics and other sciences. In the last decade, several approaches to fractional variational calculus were proposed by using different notions of fractional derivatives and integrals. Although the literature of the fractional calculus of variations is already vast, much remains to be done in obtaining necessary and sufficient conditions for the optimization of fractional variational functionals, existence and regularity of solutions. Regarding necessary optimality conditions, all works available in the literature concern the derivation of first-order fractional conditions of Euler-Lagrange type. In this work we obtain a Legendre second-order necessary optimality condition for weak extremizers of a variational functional that depends on fractional derivatives.

math.OC↗

arXiv2012

The DuBois-Reymond Fundamental Lemma of the Fractional Calculus of Variations and an Euler-Lagrange Equation Involving only Derivatives of Caputo

Derivatives and integrals of non-integer order were introduced more than three centuries ago, but only recently gained more attention due to their application on nonlocal phenomena. In this context, the Caputo derivatives are the most popular approach to fractional calculus among physicists, since differential equations involving Caputo derivatives require regular boundary conditions. Motivated by several applications in physics and other sciences, the fractional calculus of variations is currently in fast development. However, all current formulations for the fractional variational calculus fail to give an Euler-Lagrange equation with only Caputo derivatives. In this work, we propose a new approach to the fractional calculus of variations by generalizing the DuBois-Reymond lemma and showing how Euler-Lagrange equations involving only Caputo derivatives can be obtained.

math.OC↗

arXiv2012

Asymmetric exclusion model with several kinds of impurities

We formulate a new integrable asymmetric exclusion process with $N-1=0,1,2,...$ kinds of impurities and with hierarchically ordered dynamics. The model we proposed displays the full spectrum of the simple asymmetric exclusion model plus new levels. The first excited state belongs to these new levels and displays unusual scaling exponents. We conjecture that, while the simple asymmetric exclusion process without impurities belongs to the KPZ universality class with dynamical exponent 3/2, our model has a scaling exponent $3/2+N-1$. In order to check the conjecture, we solve numerically the Bethe equation with N=3 and N=4 for the totally asymmetric diffusion and found the dynamical exponents 7/2 and 9/2 in these cases.

cond-mat.stat-mech↗

arXiv2010

Asymmetric exclusion model with impurities

An integrable asymmetric exclusion process with impurities is formulated. The model displays the full spectrum of the stochastic asymmetric XXZ chain plus new levels. We derive the Bethe equations and calculate the spectral gap for the totally asymmetric diffusion at half filling. While the standard asymmetric exclusion process without impurities belongs to the KPZ universality class with a exponent 3/2, our model has a scaling exponent 5/2.

cond-mat.stat-mech↗

arXiv2007

Exactly solvable interacting vertex models

We introduce and solvev a special family of integrable interacting vertex models that generalizes the well known six-vertex model. In addition to the usual nearest-neighbor interactions among the vertices, there exist extra hard-core interactions among pair of vertices at larger distances.The associated row-to-row transfer matrices are diagonalized by using the recently introduced matrix product {\it ansatz}. Similarly as the relation of the six-vertex model with the XXZ quantum chain, the row-to-row transfer matrices of these new models are also the generating functions of an infinite set of commuting conserved charges. Among these charges we identify the integrable generalization of the XXZ chain that contains hard-core exclusion interactions among the spins. These quantum chains already appeared in the literature. The present paper explains their integrability.

cond-mat.stat-mech↗