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
$$L = e^{\gamma t} (\frac{m\dot{q}^2}{2} - \frac{kq^2}{2})$$$$L = e^{\gamma t} \left(\frac{m\dot{q}^2}{2} - \frac{kq^2}{2}\right) \, .$$
How would you describe the system? Are there any constants of motion? Suppose a point transformation is made of the form
$$ s = e^\frac{\gamma t q}{2}$$$$ s = e^\frac{\gamma t q}{2} \, .$$
What is the effective Lagrangian in terms of s$s$? Find the equation of motion for s$s$. What do these results say about the conserved quantities for the system.?
$Solution:$Solution:
Making all calculations...the equations of motion for s$s$ are: $$e^{\gamma t}(-{kq} - m\ddot{q} - \gamma m \dot{q})= 0 \\$$ $$s(m\gamma^2 -4k)=4m\ddot{s}$$\begin{align} e^{\gamma t}(-{kq} - m\ddot{q} - \gamma m \dot{q}) &= 0 \\ s (m \gamma^2 -4k) &= 4 m \ddot{s} \, . \end{align}
What do these results say about the conserved quantities for the system? That's what I 'd like to know.
