Lagrangian of 1D Motion: Finding Particle Coordinate x at Time t

[astro2cosmos]

i have L of particle m in 1D motion, but how i can find the coordinate of particle x at time t?

 

[UltrafastPED]

Now plug the Lagrangian into the Euler-Lagrange equation:
Equation [3]: http://mathworld.wolfram.com/Euler-LagrangeDifferentialEquation.html

This gives you a differential equation in f(x); you must solve the differential equation to find f(x).