Is the Lagrangian Invariant or Variant in a Coordinate System Shift?

[shounakbhatta]

Hello,

Is Lagrangian invariant?

I am in a conversation, where one is saying that:

"Shifting the coordinate system changes the value of the potential energy with respect to the same reference level, that's why the Lagrangian changes"

While the other:

"Shifting the coordinate system, doesn't shift the zero reference level of potential for the system..
The reference level of potential is fixed for the system once it is decided upon."

So, is Lagrangian variant or invariant?

Please help.

Thanks.

 

[maajdl]

The Lagrangian has the symmetries of the system that it represents.

 

[shounakbhatta]

In the case can we consider that is it variant or invariant?

Also, is it a scalar or a vector product?

 

[BruceW]

it depends what the Lagrangian is. some Lagrangian equations will be variant and some will be invariant.

edit: the definition of invariant, is that if you make some change, then the Lagrangian will still look the same. So if you shift the coordinate system but the Lagrangian still looks the same, this means the Lagrangian is invariant with respect to a coordinate shift.

 

[shounakbhatta]

If it is a variant in a equation then it cannot be scalar?

Is it that it becomes scalar as well as a vector?