Lagrangian remains invariant under addition

[preet0283]

what is the reason that the lagrangian remains invariant under addition of an arbtrary function of time?

 

[rayveldkamp]

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

 

[MalleusScientiarum]

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.