On Geodesic Behavior of Some Special Curves

Abstract

In this paper, geometric structures on an open subset D ⊆ R 2 are investigated such that the graphs associated with the solutions of some special functions to become geodesics. More precisely, we determine the Riemannian metric g such that Bessel (Hermite, harmonic oscillator, Legendre and Chebyshev) ordinary differential equation (ODE) is identified with the geodesic ODEs produced by the Riemannian metric g . The technique is based on the Lagrangian (the energy of the curve) L = 1 2 ‖ x ˙ ( t ) ‖ 2 , the associated Euler–Lagrange ODEs and their identification with the considered special ODEs.

Keywords

• auto-parallel curve
• geodesic
• Euler–Lagrange equations
• Lagrangian
• special functions