Is the Action Integral fundamental

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    Fundamental Integral
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Discussion Overview

The discussion revolves around the significance of the action integral in physics, particularly its relationship to the Euler-Lagrange equations and its applicability across various physical scenarios, including classical mechanics and string theory. Participants explore the foundational role of the action integral and its implications for boundary conditions and differential equations.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions the necessity of the action integral in physics and its connection to underlying geometry, suggesting that it may be fundamental due to its role in satisfying the Euler-Lagrange equations across different dimensions.
  • Another participant simplifies the Euler-Lagrange equation to a statement about forces, indicating a desire to verify the truth of this simplification.
  • A different participant raises the question of whether there are any physical situations that cannot be described using Lagrangian mechanics and the action integral, highlighting the universality of these concepts.
  • One participant recalls a potential limitation in deriving the Euler-Lagrange equations when the Hamiltonian exhibits explicit time dependence, although they express uncertainty about this point.

Areas of Agreement / Disagreement

The discussion contains multiple competing views regarding the fundamental nature of the action integral and its applicability. There is no consensus on whether all physical situations can be described by Lagrangian mechanics or the implications of time dependence in the Hamiltonian.

Contextual Notes

Participants express uncertainty about specific conditions under which the Euler-Lagrange equations can be derived, particularly concerning time dependence in the Hamiltonian. The discussion also touches on the relationship between boundary conditions and the Euler-Lagrange equation without resolving these complexities.

Mike2
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What is it about the action integral in physics? Symmetries in the action integral leads to the Euler-lagrange equations. And it seems every physical situation is described by equations that must satisfy the differential equation of the Euler-lagrange.

This Euler-lagrange equation must be satisfied in 3D and even in the 4D of relativity. Even if the number of parameters increase it must be satisfied over lines and surfaces of classical string theory. So what makes this Action integral so necessary to physics? Is there some underlying geometry that necessitate the Action Integral?

Can the various functions of physical situations be considered as a type of boundary conditions of the Euler-lagrange differential equation? For example, the momentum is the derivative of the Lagrangian which is a function derived from the particular physical situation, but it is also a specified first order diff eq to the second order diff eq of the Euler-lagrange eq. Isn't this a boundary value problem for a second order diff eq?
 
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(Force)-(The Same Force)=0 To me, that is all what the Euler-Lagrange equation is all about.
Sorry, I know this doesn't help, just wanted to check if that statement is kind of true.
 
So the question is: Are there ANY physical situation that can not be described with Lagrangian mechanics, by the variation of the Action integral being zero, which leads to an Euler-lagrange equation?
 
If I remember correctly, the derivation of the EUler Lagrange equations cannot be done when the Hamiltonian has an explicit time dependence in it...
Not sure though
 

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