Fractional calculus provides powerful tools for modeling nonlocality, dissipative systems, and, when defined in the time representation, provides an interesting memory effect in mathematical physics. In this paper, we review four standard fractional approaches: the Riemann–Liouville, Gerasimov–Caputo, Grünwald–Letnikov, and Riesz formulations. We present their definitions, basic properties, Weyl–Marchaud, and physical interpretations. We also give a brief review of related operators that have been used recently in applications but have received less attention in the physical literature: the fractional Laplacian, conformable derivatives, and the Fractional Action-Like Variational Approach (FALVA) for variational principles with fractional action weights. Our emphasis is on how these operators are, and can be, applied in physical problems rather than on exhaustive coverage of the field. This review is intended as an accessible introduction for physicists working in diverse areas interested in fractional calculus and fractional methods. For deeper technical or domain-specific treatments, readers are encouraged to consult the works in the corresponding fields, for which the bibliography suggests a starting point.

So far, it is not well known how to deal with dissipative systems. There are many paths of investigation in the literature and none of them present a systematic and general procedure to tackle the problem. On the other hand, it is well known that the fractional formalism is a powerful alternative when treating dissipative problems. In this paper, we propose a detailed way of attacking the issue using fractional calculus to construct an extension of the Dirac brackets in order to carry out the quantization of nonconservative theories through the standard canonical way. We believe that, by using the extended Dirac bracket definition, it will be possible to analyze more deeply gauge theories starting with second-class systems.

One has recently presented an extension of the standard variational calculus to include the presence of deformed derivatives, both in the Lagrangian of systems of particles and in the Lagrangian density of field-theoretic models. Classical Euler-Lagrange equations and the Hamiltonian formalism have been reassessed in this approach. Whenever applied to a number of physical systems, the resulting dynamical equations come out to be the correct ones found in the literature, especially with mass-dependent and with nonlinear equations for classical and quantum-mechanical systems. In the present contribution, one extends the variational approach, including a piecewise form of deformed derivatives to study higher-order dissipative systems and to obtain, as an option, deformed equations as well. Applications to concrete situations are contemplated, such as an accelerated point charge—this is the problem of the Abraham-Lorentz-Dirac force—stochastic dynamics like the Langevin, the advection-convection-reaction and Fokker-Planck equations, the Korteweg-de Vries equation, the Landau-Lifshitz-Gilbert problem, and the Caldirola-Kanai Hamiltonian and heat transfer equation of the Fourier and non-Fourier types. By considering these different examples, it is shown that the formulation proposed in this paper may be a simple, but promising, path for dealing, for example, with dissipative, nonlinear, stochastic systems and the anomalous heat transfer problem, by adopting a variational approach.

We construct a fractional Laplacian spinning particle model in an external electromagnetic field that generalizes a standard relativistic spinning particle model without anti-commuting spin variables. The one-dimensional fractional Laplacian in world-line variable $\lambda$ governs the kinetic energy that is non-local in $\lambda$. The interaction between the particle’s charge and the electromagnetic four-potential is non-local in $\lambda$, while the interaction between the particle’s spin tensor and the electromagnetic field is standard. By applying the variational principle, we obtain the equations of motion for particle coordinates. We solve analytically the equations of motion in two particular cases: the constant electric and magnetic field. For more complex field configurations, the equations are, in general, non-local and non-linear. By making the assumption of a much weaker interaction term between the charge and four-potential compared with the interaction between spinning degrees of freedom and the electromagnetic field, we obtain approximate analytical solutions in the case of a quadratic electromagnetic potential.

We construct the conformable actuated pendulum model in the conformable Lagrangian formalism. We solve the equations of motion in the absence of force and in the case of a specific force resulting from torques which generalizes a well known mechanical model. Our study shows that the conformable model captures essential information about the physical system encoded in the parameters which depend on the conformability factor α. This dependence can describe internal variations such as viscous friction, transmission or environmental effects. We solve the equations of motion analytically for α=1/2 and by Frobenius’ method for α≠1/2.

We construct a class of knot solutions of the time-dependent gravitoelectromagnetic (GEM) equations in vacuum in the linearized gravity approximation by analogy with the Rañada–Hopf fields. For these solutions, the dual metric tensors of the bi-metric geometry of the gravitational vacuum with knot perturbations are given and the geodesic equation as a function of two complex parameters of the time-dependent GEM knots are calculated. By taking stationary potentials, which formally amount to particularizing to time-independent GEM equations, we obtain a set of stationary fields subjected to constraints from the time-dependent GEM knots. Finally, the Landau–Lifshitz pseudo-tensor and a scalar invariant of the static fields are computed.