End point information in lagrangain variation principle

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Discussion Overview

The discussion revolves around the Lagrangian variation principle, specifically addressing the conditions under which the action is minimized for a free particle. Participants explore the differences in requirements for initial and endpoint information in two hypothetical scenarios: one where only initial conditions are known and another where both initial and endpoint conditions are fixed.

Discussion Character

  • Exploratory
  • Debate/contested

Main Points Raised

  • One participant questions why, in a scenario using the Lagrangian variation principle, both initial and endpoint conditions must be fixed to determine the true path, contrasting it with a scenario where only initial conditions are sufficient.
  • Another participant suggests that the least action theorem is used to find conditions for the actual path, which leads to the same ordinary differential equation (ODE) that can be solved with a single set of conditions, whether initial or boundary.
  • A further inquiry is made about the physical reasoning behind fixing endpoint variations, questioning whether it is to eliminate boundary terms or if there is another justification.
  • A participant emphasizes that the principle of least action determines the actual path between two points in phase space, which inherently sets deviations at the beginning and end to zero.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of fixing endpoint conditions in the context of the Lagrangian variation principle. There is no consensus on whether this requirement is fundamentally necessary or if it can be understood differently.

Contextual Notes

The discussion highlights potential misconceptions regarding the application of the least action principle and the nature of boundary conditions in variational problems. Participants have not fully resolved the implications of fixing endpoints versus initial conditions.

Ron19932017
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In lagrangian variation we are trying to minimize the action
S = ∫t2t1 L dt.

Consider a simple case of free particle.

Imagine In a world that everyone one only knows how to solve ODE, Using euler lagrange equation, one has
d2x/dt2 = 0 , give that we know the initial position of particle in the phase space,
the people can solve for the motion.

Now imagine in a world that everyone only know variation principle. (They have some ways to measure action in every possible path and thus find out the least action one). They need to vary the path while KEEPING BOTH INITIAL point and end point fixed in the phase space. Then they can vary the path and find out the true one.

My question is, why in the first kind of world people only need to know about initial position in phase space but in the second kind of world people must know about the ending position in the phase space too ?

This "inconsistency of information" bothers me a lot. I appreciate anyone's help in explain or pointing out my misconception. Thanks.
 
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Ron19932017 said:
They need to vary the path while KEEPING BOTH INITIAL point and end point fixed in the phase space
I think you have a misconception here: this least action theorem is only used to find conditions for the actual path. These conditions lead to the same ODE , which can then be solved to find the path when given a single set of conditions -- be it initial, boundary or whatever.
 
BvU said:
I think you have a misconception here: this least action theorem is only used to find conditions for the actual path. These conditions lead to the same ODE , which can then be solved to find the path when given a single set of conditions -- be it initial, boundary or whatever.
thanks for your reply. Howvere I still don't understand why we require the end points variation to be fixed.
Is there any physical reason behind to do it?
Or we just want to get rid of the boundary terms?
 
Forgot to welcome you ! Hello Ron, :welcome:

Don't know how to make this easier: principle of least action helps determine the actual path between two points in phase space. That alone sets the deviations at begin and end to zero.

Link to least action principle or to d'Alembert[/PLAIN] principle help ?
 
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