Related papers: Fractional Generalization of Gradient and Hamilton…
A generalization of classical mechanics is obtained from a complex parametrization of the phase space. The formalism supports complex Hamiltonian functions describing non-conservative classical mechanical systems. A quantization scheme that…
Fractional calculus is a generalization of classical theories of integration and differentiation to arbitrary order (i.e., real or complex numbers). In the last two decades, this new mathematical modeling approach has been widely used to…
In two recent papers necessary and sufficient conditions for a given system of second-order ordinary differential equations to be of Lagrangian form with additional dissipative forces were derived. We point out that these conditions are not…
In this article, the existence and uniqueness about the solution for a class of stochastic fractional-order differential equation systems are investigated, where the fractional derivative is described in Caputo sense. The fractional…
A natural consequence of the fractional calculus is its extension to a matrix order of differentiation and integration. A matrix-order derivative definition and a matrix-order integration arise from the generalization of the gamma function…
Starting from kicked equations of motion with derivatives of non-integer orders, we obtain "fractional" discrete maps. These maps are generalizations of well-known universal, standard, dissipative, kicked damped rotator maps. The main…
Using Dirac's approach to constrained dynamics, the Hamiltonian formulation of regular higher order Lagrangians is developed. The conventional description of such systems due to Ostrogradsky is recovered. However, unlike the latter, the…
The Riemann-Liouville fractional integrals and derivatives are generalized for cases when fractional exponent $d$ are functions of space and times coordinates (i.e. $d=d({\bf r}(t),t)$).
In this paper we consider the gravitational field of fractal distribution of particles. To describe fractal distribution, we use the fractional integrals. The fractional integrals are considered as approximations of integrals on fractals.…
It is shown that a given non-autonomous system of two first-order ordinary differential equations can be expressed in Hamiltonian form. The derivation presented here allow us to obtain previously known results such as the infinite number of…
In this PhD thesis we introduce a generalized fractional calculus of variations. We consider variational problems containing generalized fractional integrals and derivatives, and study them using standard (indirect) and direct methods. In…
We present the basic formulation of Hamilton dynamics in complex phase space. We extend the Hamilton's function by including the imaginary part and find out the corresponding Hamilton's canonical equation of motion. Example of simple…
In this study the general formula for differential and integral operations of fractional calculus via fractal operators by the method of cumulative diminution and cumulative growth is obtained. The under lying mechanism in the success of…
The aim of this paper is to provide a fractional generalization of the Gompertz law via a Caputo-like definition of fractional derivative of a function with respect to another function. In particular, we observe that the model presented…
Refined are the known descriptions of particle behavior with the help of Hamilton function in the phase space of coordinates and their multiple derivatives. This entails existing of circumstances when at closer distances gravitational…
Fractional gradient descent has been studied extensively, with a focus on its ability to extend traditional gradient descent methods by incorporating fractional-order derivatives. This approach allows for more flexibility in navigating…
We develop physically admissible lattice models in the harmonic approximation which define by Hamilton's variational principle fractional Laplacian matrices of the forms of power law matrix functions on the n -dimensional periodic and…
We present both the Lagrangian and Hamiltonian procedures for treating higher-order equations of motion for mechanical models by adopting the Riemann-Liouville Fractional integral to describe their action. We point out and discuss its…
We derive the Helmholtz theorem for Hamiltonian systems defined on time scales in the context of nonshifted calculus of variations which encompass the discrete and continuous case. Precisely, we give a theorem characterizing first order…
A model system is considered where two dimensional electrons are confined by a harmonic potential in one direction, and are free in the other direction. Ground state in strong magnetic fields is investigated through numerical…