Explicitly dependent on time Lagrangian

[Magister]

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

$$L = \frac{\alpha}{2} (q^\prime + qbe^{-\alpha t})^2-q^2 \frac{ab}{2} e^{-\alpha t} (\alpha +b e^{-\alpha t})- \frac{k q^2}{2}$$

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

 

[Mentz114]

Is there a transformation $$q ->\bar{q }= qf(\alpha + be^{-\alpha t})$$ that puts the explicit time dependence into q ?

 

[charng]

I understand your frustration in trying to turn an explicitly time-dependent Lagrangian into one that is not explicitly dependent on time. However, there are methods that can be used to achieve this. One approach is to use a transformation of variables, such as a canonical transformation, to eliminate the explicit time dependence in the Lagrangian. This can involve introducing new variables or changing the form of the existing ones. Another approach is to use a redefinition of the Lagrangian parameters, such as changing the constants or coefficients in the Lagrangian to remove the time dependence. Both of these methods can be complex and require careful consideration, but they can be effective in transforming the Lagrangian into a form that is not explicitly dependent on time. It may also be helpful to consult with colleagues or reference materials for additional guidance in solving this problem. Keep persevering and you will find a satisfactory solution.