Fractional Euler Lagrange Equations for Irregular Lagrangian with Holonomic Constraints

Ola A Jarab’Ah

doi:10.4236/jmp.2018.98105

Abstract

In this paper the fractional Euler Lagrange equations for irregular Lagrangian with holonomic constraints have been presented. The equations of motion are obtained using fractional Euler Lagrange equations in a similar manner to the usual mechanics. The results of fractional calculus reduce to those obtained from classical calculus (the standard Euler Lagrange equations) when γ→0 and α, β are equal unity only. Two problems are considered to demonstrate the application of the formalism.

Highlights

The Euler Lagrange equations and Hamilton’s principle form the basis of Lagrangian or Hamiltonian mechanics
The results of fractional calculus reduce to those obtained from classical calculus when γ → 0 and α, β are equal unity only
The study of holonomic constrained systems is discussed in most references of classical mechanics [1] [2]; these systems describe dynamic systems with constraints depend only on the generalized coordinates qi (t )

Summary

Introduction

The Euler Lagrange equations and Hamilton’s principle form the basis of Lagrangian or Hamiltonian mechanics. Euler Lagrange equations for holonomic constrained systems with regular Lagrangian have been presented by Hasan [14] using the fractional variationl problems. In the present paper as a continuation of Jarab’ah work [15] the fractional Euler Lagrange equations are used to obtain the equations of motion for irregular Lagrangian with holonomic constraints, it seems that there are several choices of fractional Lagrangian giving the same classical limit, in other words the same classical Lagrangian. This paper is organized as follows: In Section 2, Euler Lagrange equations formulation for Irregular Lagrangian with holonomic constraints is reviewed briefly. 4. Fractional Euler Lagrange Equations for Irregular Lagrangian with Holonomic Constraints. − d dt and Equation (9) reduces to the standard Euler Lagrange equation for holonomic constraints

Examples

Conclusion