- #1
punkimedes
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Homework Statement
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^{γt} (\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^{γt/2}q##. 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.
Homework Equations
Lagrange-euler equation ##\frac{\partial L}{\partial q} - \frac{d}{dt}\frac{\partial L}{\partial \dot{q}} = 0##
The Attempt at a Solution
Using the lagrange-euler equation, I came up with one equation of motion ##m\ddot{q}+mγ\dot{q}+kq = 0##
Is this correct? I'm not sure where to go from here. Can I solve this for ##q## like a normal second order differential equation? Can I treat ##γ## as a constant knowing nothing about it? If so, the factoring doesn't seem to come out right. (let ##y=e^{rt}##, ##my^2+mγy+k = 0##, then ?)
Also, I'm not really sure what is meant by a point transformation. Does that simply mean replace that factor in the original lagrangian?