Chapter 4.P4, Problem 2P

Problems $1$ through $3$ are concerned with one degree-of-freedom systems.

A Spring-Mass System. A mass $m$ supported by a frictionless horizontal surface is attached to a spring that is, in turn, attached to a fixed support. Assuming that motion is restricted to one direction, let $x\left(t\right)$ represent the displacement of the mass from its equilibrium position. This position corresponds to the spring being at its rest length. If the spring is compressed, then $x<0$ and the mass is left of its equilibrium position. If the spring is elongated, then $x>0$ and the mass is right of its equilibrium position. Assume that the potential energy stored in the spring is given by $k{x}^2/2$, where the parameter $k$ is the spring constant of the spring. Find the action integral and Euler-Lagrange equation for $x\left(t\right)$.