End point information in lagrangain variation principle

[Ron19932017]

In lagrangian variation we are trying to minimize the action
S = ∫^t2_t1 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.

 

[BvU]

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.

 

[Ron19932017]

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?

 

[BvU]

Forgot to welcome you ! Hello Ron,

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 ?