Principle of Least Action help me clear it up

[Master J]

I've been reading Landau and Lifgarbagez's Mechanics and have some issues I need clearing up, so I hope folk here can help :)


It is stated that the Principle of Least Action is not always valid for the entire path of a system, but only for a sufficiently small segment...what does this mean?

I interpret this as saying:

"the action will not always be minimized (altho it usually is in most cases), but it may take on a maximum value. What is important is that it is an extremum. However, if we look at a sufficiently small part of ANY path, it will always be a minimum, but the entire path may not be" ...is this correct?

 

[atyy]

https://www.physicsforums.com/showthread.php?t=505009?

 

[jjustinn]

It is stated that the Principle of Least Action is not always valid for the entire path of a system, but only for a sufficiently small segment...what does this mean?

The easiest-to-visualize example I've come across is geodesics ('straight lines') on the surface of a sphere; all geodesics satisfy "minimal action", but they're only guaranteed to be an actual minimum if they're "sufficiently short". For example, there are two geodesics connecting Los Angeles, CA to New York, NY on the surface of the Earth -- one going across mainland USA, and the other going around the other side of the world. Both are "minimal" in the sense of an action principle (by definition -- they're "straight lines" / geodesics), but obviously only one is a truly minimal path. Now if you were restricted to choosing paths that were shorter than one half the circumference of the Earth, you would be guaranteed to have a minimum -- due to peculiarities of a sphere.

Hopefully I got all of the facts right there, and it was clear enough to at least get the general idea across...if I failed on either front, hopefully someone will correct the record.

 

[mathfeel]

The easiest-to-visualize example I've come across is geodesics ('straight lines') on the surface of a sphere; all geodesics satisfy "minimal action", but they're only guaranteed to be an actual minimum if they're "sufficiently short". For example, there are two geodesics connecting Los Angeles, CA to New York, NY on the surface of the Earth -- one going across mainland USA, and the other going around the other side of the world. Both are "minimal" in the sense of an action principle (by definition -- they're "straight lines" / geodesics), but obviously only one is a truly minimal path. Now if you were restricted to choosing paths that were shorter than one half the circumference of the Earth, you would be guaranteed to have a minimum -- due to peculiarities of a sphere.

Hopefully I got all of the facts right there, and it was clear enough to at least get the general idea across...if I failed on either front, hopefully someone will correct the record.

In this example, while the continental path is a global minimum, both paths are local minimum in the functional space of all LA-NY paths: it is stationary against small variation of path and in fact has a positive second order correction. Is this what L&L had in mind?

We can all agree that geodesics are locally straight.

 

[PJC01]

Yes, that is correct. The Principle of Least Action states that a system will follow a path that minimizes its action (a measure of energy and time) between two points. However, this does not necessarily mean that the entire path will be the absolute minimum. There may be certain points along the path where the action is not minimized, but as long as the overall action is minimized between the initial and final points, the system will follow that path.

In other words, the Principle of Least Action is a local principle, meaning it applies to small segments of a system's path rather than the entire path. This is because in complex systems, such as those described in Landau and Lifshitz's Mechanics, the total action may not be minimized along the entire path, but it will be minimized for a small enough segment of the path. This allows for a more accurate description of the system's behavior.

I hope this helps clarify the concept of the Principle of Least Action for you. If you have any further questions, please don't hesitate to ask.