End point information in lagrangain variation principle

In summary, in lagrangian variation, we are trying to minimize the action S = ∫t2t1 L dt. This involves finding the least action path between two points in phase space. In a world where people only know how to solve ODEs, this can be done by using the Euler-Lagrange equation. However, in a world where people only know the variation principle, they must vary the path while keeping both the initial and ending points fixed in the phase space. This may seem like an inconsistency, but the conditions for the actual path can still be found using the same ODE. The requirement to fix both the initial and ending points may be to get rid of boundary terms. The principle of least action helps
  • #1
Ron19932017
32
3
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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  • #2
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.
 
  • #3
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?
 
  • #4
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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FAQ: End point information in lagrangain variation principle

1. What is the Lagrangian variation principle?

The Lagrangian variation principle is a fundamental principle in classical mechanics that states that the true path of a physical system is the one that minimizes the action integral, which is the integral of the Lagrangian function over time.

2. What is the significance of the end point information in Lagrangian variation principle?

The end point information in the Lagrangian variation principle is crucial because it allows us to determine the equations of motion for a physical system. By specifying the initial and final positions and velocities of a system, we can use the Lagrangian variation principle to find the path that the system will follow.

3. How is the end point information used in the Lagrangian variation principle?

The end point information is used to set up the boundary conditions for the system. These conditions are then used to derive the equations of motion for the system by minimizing the action integral.

4. Can the Lagrangian variation principle be applied to all physical systems?

Yes, the Lagrangian variation principle can be applied to any physical system, as long as the system can be described by a Lagrangian function. This includes both classical and quantum mechanical systems.

5. What are the advantages of using the Lagrangian variation principle?

The Lagrangian variation principle offers a more elegant and general approach to solving problems in classical mechanics compared to Newton's laws of motion. It also allows for the incorporation of constraints and provides a systematic way to derive the equations of motion for a system.

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