The Hamilton formalism with fractional derivatives

Abstract

Recently the traditional calculus of variations has been extended to be applicable for systems containing fractional derivatives. In this paper the passage from the Lagrangian containing fractional derivatives to the Hamiltonian is achieved. The Hamilton's equations of motion are obtained in a similar manner to the usual mechanics. In addition, the classical fields with fractional derivatives are investigated using Hamiltonian formalism. Two discrete problems and one continuous are considered to demonstrate the application of the formalism, the results are obtained to be in exact agreement with Agrawal's formalism.

Fractional Newtonian mechanics

“…It has been shown that Lagrangians involving fractional time derivatives lead to equations of motion with non-conservative classical forces. Recently, several researchers have explored this area and given new insight into this problem [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]. A fractional calculus with variations which deal with problems containing both the left-and right derivatives has been developed [40].…”

Time-fractional extensions of the Liouville and Zwanzig equations

“…It is necessary to note that the fractional generalizations of the Hamiltonian formalism have been actively developed in recent years [32][33][34][35][36][37]. In particular in [35][36][37], the fractional Hamilton's equations of motion have been obtained from the variational problems for the Lagrangians depending on different types of fractional derivatives of coordinates.…”

Constant curvature coefficients and exact solutions in fractional gravity and geometric mechanics

“…As it is known the fractional calculus which deals with derivative and integrals of arbitrary orders [22][23][24][25][26] is an emerging field which has many applications in various fields of science and engineering [26][27][28][30][31][32][33][34][35].…”