Physical Interpretation of point transformation invariance of the Lagrangian

[anton01]

Homework Statement

The problem asked us to show that the Euler-Lagrange's equations are invariant under a point transformation q_{i}=q_{i}(s_{1},...,s_{n},t), i=1...n. Give a physical interpretation.

Homework Equations

\frac{d}{dt}(\frac{\partial L}{\partial \dot{s_{j}}})=\frac{\partial L}{\partial s_{j}}

The Attempt at a Solution

I proved the invariance.
I am stumped with the physical interpretation. Except for the fact that the E-L equations are invariant when we change coordinates pointwise, I don't see any other physical interpretation. But this answer seems just repeating their question.

 

[voko]

Are coordinates something physical to begin with?

 

[anton01]

Are coordinates something physical to begin with?

No they are not. They are something we make in order to do the calculations.
It was just a wild guess really.

 

[voko]

That's what one should expect physically. Now you have proved that the Lagrangian formalism does not require any special coordinates, they all work. What does that mean about the formalism itself?

 

[anton01]

This means that the Lagrangian is independent of the coordinate system. And this makes sense, because a coordinate system is nothing physical and is arbitrary.
So, this means that the Lagrangian is universal? In other words, it works for any coordinate system.