Understanding the Euler Lagrange Equation and Its Boundary Condition

Click For Summary

Discussion Overview

The discussion centers around the Euler Lagrange equation and its boundary conditions, focusing on the derivation process and the implications of different types of boundary conditions (fixed and free). Participants explore the role of variations in the functional and how they relate to boundary conditions.

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant expresses confusion about the boundary condition that requires a value to be zero.
  • Another participant suggests that this may relate to the variation being zero at the endpoints.
  • A later reply confirms the connection between the boundary condition and the variation at the endpoints.
  • One participant argues that boundary conditions are not necessary to derive the Euler Lagrange equation, stating that all variations must yield zero variation of the functional, particularly those with fixed endpoints.
  • Another participant introduces the idea that if boundary conditions are not fixed, considering variations without fixed endpoints will lead to boundary conditions, referred to as free boundary conditions.
  • One participant challenges the notion that variations must be zero at the endpoints, suggesting that boundary conditions may arise from surface terms in the action.

Areas of Agreement / Disagreement

Participants express differing views on the necessity and nature of boundary conditions in relation to the Euler Lagrange equation. There is no consensus on whether variations must be zero at the endpoints or how boundary conditions arise.

Contextual Notes

Some assumptions about the nature of boundary conditions and the types of variations considered remain unresolved. The discussion includes references to fixed and free boundary conditions without a clear resolution on their implications.

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).
 

Similar threads

Undergrad Standard designation for generalization of Euler-Lagrange?
  • · Replies 5 ·
Replies
5
Views
2K
Graduate Lagrangian to the Euler-Lagrange equation
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Action in Lagrangian Mechanics
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Any inflexion-point solutions to Euler-Lagrange equation?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Linearization of Lagrange equations
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Landau's inertial frame logic
  • · Replies 1 ·
Replies
1
Views
1K
Graduate Proving Euler Lagrange equations
  • · Replies 2 ·
Replies
2
Views
1K
Graduate H-Theorem and Lagrange multipliers
  • · Replies 7 ·
Replies
7
Views
2K
Graduate What is the Spin Connection Equation for Gauging the Poincare Group?
  • · Replies 2 ·
Replies
2
Views
2K
Covariant Euler-Lagrange Computation
  • · Replies 6 ·
Replies
6
Views
3K