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

$$L = e^{\gamma t} (\frac{m\dot{q}^2}{2} - \frac{kq^2}{2})$$


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

$$ s = e^\frac{\gamma t q}{2}$$

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.

$Solution:$

Making all calculations...the equations of motion for s are:
$$e^{\gamma t}(-{kq} - m\ddot{q} - \gamma m \dot{q})= 0 \\$$
$$s(m\gamma^2 -4k)=4m\ddot{s}$$

What do these results say about the conserved quantities for the system? That's what I 'd like to know.