The invariance of Lagrange's equations with a given time

[GAclifton]

The invariance of Lagrange's equations with a given "time"

Homework Statement

What is the change in the Lagrangian in order that the Lagrangian equations of motion retain their form under the transformation to new coordinates and "time" give by:

q = q(Q, $$\tau$$)
t = t(Q, $$\tau$$)

Homework Equations

The Lagrange equations of motion.

*That tau is not supposed to be a superscript of anything. I tried to write the LaTex code myself and it didn't work. It's just supposed to be regular lower case tau.

The Attempt at a Solution

I have shown that the Lagrange equations of motion are invariant under a coordinate transformation of the same time, but I can't get this one to workout because I don't know how far I need to take the partials.

 

[diazona]

Your LaTeX seems fine to me...

I'm not sure I entirely understand what you're asking. Can you post some of your work, specifically the part where you ran into problems?

 

[GAclifton]

It has been shown that for q = q(Q, t), the lagrangian transforms invariantly. You assume a good transformation to Q = Q(q, t). You go through the formalism and bam, the form of the lagrangian equations of motion are the same in the Q frame and the q frame.

The difference here is that now we have q = q(Q, $$\tau$$) and t = t(Q, $$\tau$$). So you have to go through the same formalism in switching to Q = Q(q, t) and$$\tau$$ =$$\tau$$(q, t). Does that help? You now have to consider partial derivatives of q wrt t and q wrt $$\tau$$ etc.

 

[diazona]

I understand what the question is asking for, but I'm still not clear on exactly what problem you're having. It would really help if you post the part of your work where you're having trouble.