Novel fractional-order Lagrangian to describe motion of beam on nanowire

Our aim in this research is to investigate the motion of a beam on an internally bent nanowire by using the fractional calculus theory. To this end, we first formulate the classical Lagrangian which is followed by the classical Euler–Lagrange equation. Then, after introducing the generalized fractional Lagrangian, the fractional Euler–Lagrange equation is provided for the motion of the considered beam on the nanowire. An efficient numerical scheme is introduced for implementation and the simulation results are reported for different fractional-order values and various initial settings. These results indicate that the fractional responses approach the classical ones as the fractional order goes to unity. In addition, the fractional Euler–Lagrange equation provides a flexible model possessing more information than the classical description — the fact that leads to a considerably better evaluation of the hidden features of the real system under investigation.