Understanding the Euler Lagrange Equation and Its Boundary Condition

Click For Summary
The discussion focuses on the boundary conditions necessary for deriving the Euler-Lagrange (EL) equation. It clarifies that fixed boundary conditions are not required to obtain the EL equation, as all variations must yield zero variation of the functional, particularly those with fixed endpoints. The conversation also touches on the concept of free boundary conditions, which arise when variations are considered without fixed endpoints. Additionally, it suggests that boundary conditions may originate from surface terms in the action. Understanding these aspects is crucial for correctly applying the Euler-Lagrange equation in variational calculus.
TimeRip496
Messages
249
Reaction score
5
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?
 
Physics news on Phys.org
Can you elaborate?
Is this related to the variation being zero at the endpoints?
 
robphy said:
Can you elaborate?
Is this related to the variation being zero at the endpoints?
Yes.
 
TimeRip496 said:
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.
 
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).
 

Thread 'Does acceleration affect impact energy vs constant velocity?'

I do not have a good working knowledge of physics yet. I tried to piece this together but after researching this, I couldn’t figure out the correct laws of physics to combine to develop a formula to answer this question. Ex. 1 - A moving object impacts a static object at a constant velocity. Ex. 2 - A moving object impacts a static object at the same velocity but is accelerating at the moment of impact. Assuming the mass of the objects is the same and the velocity at the moment of impact...
View full post »

Similar threads

Proving Euler Lagrange equations
  • · Replies 2 ·
Replies
2
Views
1K
Covariant Euler-Lagrange Computation
  • · Replies 6 ·
Replies
6
Views
3K
Lagrange's Equations: Know When to Use Each
  • · Replies 1 ·
Replies
1
Views
1K
Graduate Find Euler-Lagrange Equations w/Given Init Pos & Vel
  • · Replies 12 ·
Replies
12
Views
2K
Undergrad Standard designation for generalization of Euler-Lagrange?
  • · Replies 5 ·
Replies
5
Views
2K
Modifying Euler-Lagrange equation to multivariable function
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Lagrangian and the Euler Lagrange equation
  • · Replies 2 ·
Replies
2
Views
1K
Graduate Euler Lagrange Equations for simple multi-body systems
  • · Replies 1 ·
Replies
1
Views
1K
High School Significance of the solution of the Euler-Lagrange equation
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Principle of Stationary Action - Intuition
  • · Replies 13 ·
Replies
13
Views
3K