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

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Discussion Overview

The discussion revolves around the invariance or variance of the Lagrangian in the context of coordinate system shifts. Participants explore whether shifting the coordinate system affects the value of the Lagrangian, particularly in relation to potential energy and reference levels.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • One participant suggests that shifting the coordinate system changes the value of potential energy, implying that the Lagrangian changes as a result.
  • Another participant argues that the reference level of potential is fixed for the system, indicating that the Lagrangian should not change with a coordinate shift.
  • It is proposed that the invariance of the Lagrangian depends on its specific form, with some Lagrangians being invariant and others being variant under coordinate transformations.
  • A later reply defines invariance as the property that the Lagrangian remains unchanged in appearance when a coordinate shift is applied.
  • There is a question raised about whether a variant Lagrangian can be considered a scalar, suggesting a potential relationship between variance and the nature of the Lagrangian as scalar or vector.

Areas of Agreement / Disagreement

Participants express differing views on whether the Lagrangian is invariant or variant under coordinate shifts, indicating that the discussion remains unresolved with multiple competing perspectives.

Contextual Notes

Participants do not clarify the specific forms of the Lagrangian being discussed, nor do they resolve the implications of potential energy reference levels on the invariance of the Lagrangian.

shounakbhatta
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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.
 
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The Lagrangian has the symmetries of the system that it represents.
 
In the case can we consider that is it variant or invariant?

Also, is it a scalar or a vector product?
 
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.
 
If it is a variant in a equation then it cannot be scalar?

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

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