Lagrangian remains invariant under addition

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The Lagrangian remains invariant under the addition of an arbitrary function of time because the equations of motion are unaffected by such additions, specifically when the function is a total time derivative. This is due to the Euler-Lagrange equations relying solely on derivatives with respect to position and velocity, rendering the added function's partial derivatives irrelevant. The discussion highlights the foundational role of this invariance in advanced concepts like contact transformations and Hamilton-Jacobi theory. By identifying suitable functions and variable changes, complex classical physics problems can be simplified significantly. Understanding this principle is crucial for deeper insights into classical mechanics.
preet0283
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what is the reason that the lagrangian remains invariant under addition of an arbtrary function of time?
 
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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
 
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.
 

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