Explicitly dependent on time Lagrangian

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SUMMARY

The discussion focuses on transforming a time-dependent Lagrangian, specifically L = (α/2)(q' + qbe^(-αt))^2 - (q^2 ab/2)e^(-αt)(α + be^(-αt)) - (kq^2/2), into a form that is not explicitly dependent on time. The user seeks methods for achieving this transformation, suggesting a potential substitution of the form q -> q̄ = qf(α + be^(-αt)). The inquiry highlights the challenge of eliminating time dependence in Lagrangian mechanics.

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I need to turn this Lagrangian in one that is not explicitly dependent on time.

[tex] L = \frac{\alpha}{2} (q^\prime + qbe^{-\alpha t})^2-q^2 \frac{ab}{2} e^{-\alpha t} (\alpha +b e^{-\alpha t})- \frac{k q^2}{2}<br /> [/tex]

I have already spent a lot of time around this problem but I am far from getting a satisfactory answer. Is there any method for doing this or is just by looking? Any help would be great! Thanks
 
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Is there a transformation [tex]q ->\bar{q }= qf(\alpha + be^{-\alpha t})[/tex] that puts the explicit time dependence into q ?
 

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