This paper aims to find the motion of a particle that is restricted to move along a catenary curve. The method used to find the dynamics of the particle , we use Lagrangian mechanics along with elliptical integrals to find solve the obtained equation of motion for time. We transform the kinetic and potential energies one generalized coordinate, which results in a short Lagrangian formulation.After applying the Euler—Lagrange equation and using lagrange multiplier which shortens it down and helps us to find the constraint force needed helps us to reach to a differential equation i.e., the equation of motion is derived, analyzing the interaction between inertial and gravitational forces. A similar process is carried upon for the particle going on a catenary under influence of gravity and gravity too. The Euler Lagrange Equation is modified for non conservative forces and a equation of motion is derived. Using Lagrangian Mechanics yield a simple and elegant method