Conserved quantities for the system, classical mechanics

Question

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 \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?

Answer 1

Since the transformation creates a Lagrangian for the system with no explicit time dependence, there is a conserved energy integral (Hamiltonian) that can be constructed in the new coordinates. The quantity is still conserved if you transform back into the original ($q$) coordinates, but in those coordinates, it is explicitly time dependent; in fact, it basically looks like the energy multiplied by an exponentially growing factor to keep it constant.