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, [tex]\tau[/tex])
t = t(Q, [tex]\tau[/tex])

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, [tex]\tau[/tex]) and t = t(Q, [tex]\tau[/tex]). So you have to go through the same formalism in switching to Q = Q(q, t) and[tex]\tau[/tex] =[tex]\tau[/tex](q, t). Does that help? You now have to consider partial derivatives of q wrt t and q wrt [tex]\tau[/tex] 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.

 

[paranormal]

As a scientist, it is important to understand that the invariance of Lagrange's equations with a given time is a fundamental principle in classical mechanics. It states that the equations of motion derived from the Lagrangian are independent of the choice of coordinates and the choice of time parameterization. This means that the form of the equations will remain the same regardless of the specific coordinates and time parameterization used.

In order to maintain this invariance, the Lagrangian must also remain unchanged under a transformation of coordinates and time. This is known as the principle of least action, which states that the physical system will follow the path that minimizes the action, which is the integral of the Lagrangian over time.

To ensure the invariance of Lagrange's equations, we must consider the transformation of both coordinates and time simultaneously. This means that the Lagrangian must be a function of both the original coordinates and time, as well as the transformed coordinates and time. In other words, the Lagrangian must be invariant under the transformation:

q = q(Q, τ)
t = t(Q, τ)

In order to determine the change in the Lagrangian, we can use the chain rule to express the Lagrangian in terms of the new coordinates and time:

L(q, t) = L(q(Q, τ), t(Q, τ))

We can then take the partial derivatives of the Lagrangian with respect to the new coordinates and time, and set them equal to zero to ensure that the Lagrangian remains unchanged:

∂L/∂Q = 0
∂L/∂τ = 0

These conditions will ensure that the Lagrangian remains invariant and the equations of motion retain their form under the given transformation of coordinates and time. This not only allows for a more general and flexible description of physical systems, but also reinforces the fundamental principle of invariance in classical mechanics.