Lagrangian remains invariant under addition

In summary, the Lagrangian remains invariant under the addition of an arbitrary function of time because the equations of motion are also invariant. This allows for the inclusion of a wider range of functions, leading to more advanced concepts like contact transformations and the Hamilton-Jacobi theory. These additions can simplify classical physics problems significantly.
  • #1
preet0283
19
0
what is the reason that the lagrangian remains invariant under addition of an arbtrary function of time?
 
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  • #2
Hi,
It is the equations of motion that are invariant under the addition of a function that is the total time derivative of some function, to the Lagrangian. Since the Euler-Lagrange equations involve derivatives with respect to position and velocity only, a partial derivative wrt to position or velocity of this added function will be zero.
Hope this helps

Ray
 
  • #3
Preet, what you just noticed is the basis for later things like contact transformations and the resulting Hamilton-Jacobi theory. It turns out that you can add a more general class of functions whose derivatives obey a certain relationship, and if you can find these functions and changes of variable then you can make any classical physics problem a piece of cake.
 

What does it mean for Lagrangian to remain invariant under addition?

It means that the Lagrangian function, which describes the dynamics of a system, remains unchanged when a constant value is added to it. This is a fundamental principle in physics, known as the principle of least action.

Why is it important for Lagrangian to remain invariant under addition?

This principle is important because it leads to the equations of motion that govern the behavior of a physical system. It also allows for the conservation of energy, momentum, and angular momentum.

How is Lagrangian affected by adding a constant value?

Adding a constant value to the Lagrangian function does not change the behavior of the system. The equations of motion and the conservation laws remain the same, as long as the added constant is the same for all points in time and space.

Can Lagrangian be invariant under other mathematical operations?

Yes, the principle of least action can be generalized to other mathematical operations, such as multiplication or differentiation. However, addition is the most commonly used operation in physics.

Is the principle of least action unique to Lagrangian?

No, similar principles exist in other areas of physics, such as Hamilton's principle of stationary action, which is based on the Hamiltonian function. However, Lagrangian remains the most widely used approach in classical mechanics.

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