How Does Adding a Total Time Derivative Affect the Lagrangian's Action?

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Homework Help Overview

The discussion revolves around the implications of adding a total time derivative to a Lagrangian in the context of classical mechanics and action principles. Participants explore the conceptual understanding of how this modification affects the equations of motion and the physical interpretation of the Lagrangian.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the mathematical proof of the invariance of equations of motion under the addition of a total time derivative. They explore examples, such as the addition of a constant to potential energy, and question the implications of this gauge freedom in the Lagrangian. Some express curiosity about the role of the function's dependence on coordinates and time only, and how this relates to the minimum action principle.

Discussion Status

The discussion is active, with participants sharing insights and examples while seeking further clarification on the conceptual implications of their findings. There is an acknowledgment of the relationship between the Lagrangian and physical quantities, with some participants expressing hope for deeper insights in future readings.

Contextual Notes

Participants are operating under the framework of classical mechanics and are referencing established texts, such as Goldstein, to further their understanding. There is an exploration of the concept of gauge invariance and its implications for the action principle.

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Lagrangian --> Lagrangian'

It is said that the equations of motion remain unchaged if one switches from a known lagrangian to a new lagrangian plus a total time derivative of a function. IIRC, the function is a function of q and t only. I have shown this to be so mathematically, but am trying to understand what it implies conceptually. Thanks.
 
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A simple example would be to add a constant to the potential energy. Here, the value of the potential energy has some arbitrariness to it, but the equations of motion are unaffected. This makes sense because it's that change in potential that affects motion and not the assigned value. The property you mentioned generalizes this further.
 
This "gauge freedom" in the lagrangian will also reveal itself to be very key. It is at the heart of Noether's thm and canonical transformations in the hamiltonian formulation.
 
Thanks. I can understand the example of the potential.
Hopefully Goldstein bears this out more in later chapters.
Any other thoughts?
 
I guess the lesson is that the Lagrangian isn't a physical quantity. But the derivatives (both with time and space) of the Lagrangian are physical quantities.

If you're familiar with the action, you know that it's the integral of the Lagrangian over time. The path of minimum action is the physically realized path of a system with that Lagrangian. This path will be unchanged when you add a total time derivative to the Lagrangian. The reason is pretty clear. You're simply adding a constant to the action for every path, so it won't change which path minimizes the action--it will only change the value of the action over that path. But, again, the action isn't a physical quantity--but the path of least action is!
 
So it sounds very much like a gauge invariance. Why can the function only depend on the coords and time? No velocity, etc. I am trying to imagine how that might change the minimum value of the action over a path.
thanks again
 

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