From Calculus of variations to Fractional Euler–Lagrange equations.

The role of calculus of variations in aerodynamics studies is related to the study of optimization problems in aircraft dynamics which include the study of the optimal trajectory using the least fuel consumption or optimum geometry in aircraft design or optimal distances of the aircraft or rocket trajectory subjects to certain conditions. In the calculus of variation, the optimal solution is obtained by the Euler-Lagrange equation. The fractional calculus is a mathematical theory for calculating derivatives and integrals where the order can be fractions or complex numbers. There are numerous applications of this theory in various fields of science and engineering today. In this paper, we have applied theory of fractional calculus to formulate the generalized term of Euler-Lagrange equation. The ubiquitous equation in mechanics and other field.