Generalized Variational Problems and Euler–Lagrange equations

This paper introduces three new operators and presents some of their properties. It defines a new class of variational problems (called Generalized Variational Problems, or GVPs) in terms of these operators and derives Euler–Lagrange equations for this class of problems. It is demonstrated that the left and the right fractional Riemann–Liouville integrals, and the left and the right fractional Riemann–Liouville, Caputo, Riesz–Riemann–Liouville and Riesz–Caputo derivatives are special cases of these operators, and they are obtained by considering a special kernel. Further, the Euler–Lagrange equations developed for functional defined in terms of the left and the right fractional Riemann–Liouville, Caputo, Riesz–Riemann–Liouville and Riesz–Caputo derivatives are special cases of the Euler–Lagrange equations developed here. Examples are considered to demonstrate the applications of the new operators and the new Euler–Lagrange equations. The concepts of adjoint differential operators and adjoint differential equations defined in terms of the new operators are introduced. A new class of generalized Lagrangian, Hamiltonian, and action principles are presented. In special cases, these formulations lead to fractional adjoint differential operators and adjoint differential equations, and fractional Lagrangian, Hamiltonian, and action principle. Thus, the new operators introduce a generalized approach to many problems in classical mechanics in general and variational calculus in particular. Possible extensions of the subject and the concepts discussed here are also outlined.