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 $$L = e^{\gamma t} \left(\frac{m\dot{q}^2}{2} - \frac{kq^2}{2}\right).$$ How would you describe the system? Are there any constants of motion? Suppose a point transformation is made of the form $$ s = e^\frac{\gamma t q}{2}.$$
- 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.
