Understanding the Euler Lagrange Equation and Its Boundary Condition

[TimeRip496]

I am trying to derive it but I am stuck at the boundary condition. What is this boundary comdition thing such that the value must be zero?

 

[robphy]

Can you elaborate?
Is this related to the variation being zero at the endpoints?

 

[TimeRip496]

Can you elaborate?
Is this related to the variation being zero at the endpoints?

Yes.

 

[Orodruin]

Yes.

You do not need the boundary conditions to get to the EL equation. In order to extremise the functional, all variations must give zero variation of the functional, in particular those variations with fixed end points.

Now, if you do not have fixed BCs, also considering variations without fixed BCs will give you BCs when requiring zero variation of the functional - so called free BCs.

 

[ChrisVer]

the variation can be zero at the endpoints, but it may not be...I think that your BCs are coming from the surface terms (the total-divergent terms in your action).