Lagrangian -> Lagrangian'

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In summary, Lagrangian --> Lagrangian' is a mathematical way of saying that the equations of motion remain unchanged if you switch from a known lagrangian to a new lagrangian plus a total time derivative of a function. This function is typically a function of q and t only, and adding a total time derivative of the Lagrangian will only change the value of the action over a path that minimizes the Lagrangian. This is because the action is a physical quantity, but the path of least action is still a physical quantity.
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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.
 
  • #3
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.
 
  • #4
Thanks. I can understand the example of the potential.
Hopefully Goldstein bears this out more in later chapters.
Any other thoughts?
 
  • #5
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!
 
  • #6
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
 

What is the Lagrangian method and how is it used in science?

The Lagrangian method is a mathematical technique used to describe the motion of a system of particles or objects. It is a powerful tool in physics and engineering, allowing scientists to model complex systems and predict their behavior.

What is the difference between the Lagrangian and Hamiltonian methods?

The Lagrangian and Hamiltonian methods are both mathematical techniques used to describe the motion of systems. However, the Lagrangian method is based on the concept of energy, while the Hamiltonian method is based on the concept of momentum. The two methods can be used interchangeably, but some systems may be better described using one method over the other.

What are the advantages of using the Lagrangian method over other methods of analyzing systems?

One advantage of using the Lagrangian method is its ability to easily incorporate constraints and forces into the equations of motion. This makes it useful for studying systems with complex interactions and multiple constraints. It also provides a more intuitive understanding of the system's behavior, as it is based on the concept of energy.

How does the Lagrangian method relate to Newton's laws of motion?

The Lagrangian method is based on the principle of least action, which is closely related to Newton's laws of motion. Both describe the behavior of a system in terms of the forces acting upon it. However, the Lagrangian method provides a more elegant and efficient way of solving for the system's motion.

What types of systems can be analyzed using the Lagrangian method?

The Lagrangian method can be applied to a wide range of systems, from simple particles to complex systems with multiple interacting objects. It is commonly used in classical mechanics, but can also be applied to other areas of science, such as electromagnetism and quantum mechanics.

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