Invariant Lagrangian or action

In summary, the invariance of the Lagrangian and the invariance of the action are not necessarily equivalent. While the Lagrangian can vary by a total derivative and still leave the action invariant, the action remains invariant only if the Lagrangian is truly invariant. This difference is important to consider when discussing symmetries in physics, as space-time symmetries and internal symmetries have different effects on the invariance of the Lagrangian and the action.
  • #1
Physiana
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"invariant" Lagrangian or action

Hello everyone,

I tried to describe my question but it seems getting too complicated and confusing to write down my thoughts in detail, so I am trying to start with the following question...

Are invariance of the Lagrangian under a transformation and invariance of the action equivalent to each other? or even the same?

Physiana

P.S.: I know it sounds stupid. But I was wondering if there is a difference in physics by demanding the Lagrangian being invariant or just transforming in a certain way. As for example for time dilatation [itex]
t \rightarrow t' = \lambda t
[/itex] the Lagrangian transforms as[itex]
L \rightarrow L'= \frac{L}{\lambda}
[/itex] which is not "exactly" invariant, although "a" is constant and probably does not really matter. So I wondered if there are more transformations that "change" the Lagrangian but leave the action invariant.
 
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  • #2


They're not quite the same, the Lagrangian can vary by a total derivative which would leave the action invariant, since it can be integrated over the boundary by stokes' theorem, and it is normally assumed fields all go to zero at infinity.

The variation of the lagrangian [itex]\delta L[/itex] contributes one term to the Noether current of that symmetry.
 
  • #3


So there is a difference between a Lagrangian being invariant and it "transforming as".

Does it say anything about the underlying symmetry, if a Lagrangian is invariant or "only" transforming as? (e.g. space-time, external, internal)

In field theories one generally speaks of invariance, right?

It is just, I have read too much these days and I ram starting to get really confused and loose what I believed to know/ understand.
 
  • #4


To specify; For space time transformations I get a [itex] \delta L [/itex], while for all the symmetries of the Standard model (QCD, GWS and chiral) the Lagrangian remains invariant. Space time symmetries are external symmetries, while QCD, GWS and chiral symmetries are internal symmetries. So can I generalize the above "observed" transformation properties of the Lagrangian to all external resp. internal symmetry transformations?

I just ask because I am writing my thesis and I am not exactly sure which words to use and it is important to me to be as clear as possible in my use of words.
 
  • #5


Physiana said:
To specify; For space time transformations I get a [itex] \delta L [/itex], while for all the symmetries of the Standard model (QCD, GWS and chiral) the Lagrangian remains invariant. Space time symmetries are external symmetries, while QCD, GWS and chiral symmetries are internal symmetries. So can I generalize the above "observed" transformation properties of the Lagrangian to all external resp. internal symmetry transformations?

I just ask because I am writing my thesis and I am not exactly sure which words to use and it is important to me to be as clear as possible in my use of words.

Under a spacetime symmetry the action is invariant if and only if

[tex]\delta \mathcal{L} + \partial_{\mu}(\delta x^{\mu} \mathcal{L}) = 0[/tex]

For internal symmetries; [itex]\delta x^{\mu} = 0[/itex], therefore, the action is invariant if and only if the Lagrangian is invariant. See post #12 in

www.physicsforums.com/showthread.php?t=172461

regards

sam
 
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  • #6


:) Thank you.
 

What is an invariant Lagrangian or action?

An invariant Lagrangian or action is a mathematical quantity used in physics to describe the dynamics of a system. It is a function that takes into account the position, velocity, and other relevant variables of a system, and can be used to derive the equations of motion for that system. The term "invariant" refers to the fact that the Lagrangian or action remains unchanged under certain transformations, such as changes in the coordinate system or the frame of reference.

What is the significance of an invariant Lagrangian or action?

The use of an invariant Lagrangian or action allows for a more elegant and unified approach to studying the dynamics of a system. It simplifies the equations of motion and allows for better predictions of the behavior of a system. Additionally, the invariance property ensures that the dynamics of a system remain the same regardless of the chosen coordinate system, making it a powerful tool for analyzing physical phenomena.

How is an invariant Lagrangian or action related to the principle of least action?

The principle of least action states that the actual path taken by a system between two points in time is the one that minimizes the action, which is the integral of the Lagrangian over time. In other words, the system will follow the path that requires the least amount of energy to travel. This is closely related to the invariance property of the Lagrangian, as the least action path will remain the same regardless of the chosen coordinate system.

What are some examples of systems described by an invariant Lagrangian or action?

Some examples include classical mechanics systems such as a pendulum or a simple harmonic oscillator, as well as more complex systems like a bouncing ball or a swinging double pendulum. In quantum mechanics, the Schrödinger equation can also be derived from an invariant action, and in general relativity, the Einstein field equations can be derived from an invariant action.

How is an invariant Lagrangian or action used in practical applications?

Invariant Lagrangians and actions are used extensively in theoretical physics, particularly in the fields of classical mechanics, quantum mechanics, and general relativity. They are also used in practical applications such as in the design of control systems for spacecraft and in the development of computational algorithms for simulating physical systems. Invariant Lagrangians and actions provide a powerful tool for understanding and predicting the behavior of complex systems in a wide range of contexts.

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