In certain situations, particularly one-dimensional systems, it is possible to incorporate...

In certain situations, particularly one-dimensional systems, it is possible to incorporate frictional effects without introducing the dissipation function. As an example, find the equations of motion for the Lagrangian

How would you describe the system? Are there any constants of motion? Suppose a point transformation is made of the form

What is the effective Lagrangian in terms of s? Find the equation of motion for s. What do these results say about the conserved quantities for the system?