Principle of Least Action - Straight Worldline on a Geodesic

[runner108]

What does it mean to say that something moves on a straight wordline in terms of the principle of least action? I know it generally means that action is minimum or stationary but since I only really know some physics from a conceptual standpoint and not a mathematical one I don't really know what this means. Does it mean for geodesic motion that action is Zero?

Another way of putting the question is, is there anyway to tell from the output of the Principle of Least Action that something is moving on a straight worldline as opposed to a curved one? Does one always reduce to 0 and the other is non-zero? Or is that wrong.

I'm trying to figure out the privileged status of movement along a geodesic in terms of the principle of least action.

 

[MaxwellsDemon]

Geodesics are just the straightest possible "lines" through curved spacetime. The principle of least action is basically like a version of Fermat's principle of least time... only for material bodies instead of light. Recall that Fermat's principle says that light always travels the path between two points that takes the least amount of time. In curved spacetime that would be a null geodesic. Material bodies without any forces acting on them move along timelike geodesics in general relativity. So the statement that it moves along a straight worldline is just saying that it follows a timelike geodesic (has no forces acting on it)...that it makes use of the principle of least action...all things do that.

 

[haushofer]

What does it mean to say that something moves on a straight wordline in terms of the principle of least action? I know it generally means that action is minimum or stationary but since I only really know some physics from a conceptual standpoint and not a mathematical one I don't really know what this means. Does it mean for geodesic motion that action is Zero?

No, the action doesn't have to be zero, but its variation.

Another way of putting the question is, is there anyway to tell from the output of the Principle of Least Action that something is moving on a straight worldline as opposed to a curved one? Does one always reduce to 0 and the other is non-zero? Or is that wrong.

I'm trying to figure out the privileged status of movement along a geodesic in terms of the principle of least action.

This depends on the action you take. For instance, without external force fields (but with the probability of gravity) you just have as the action of a particle the length of the worldline. This gives you the geodesic equation. In the presence of, say, electromagnetic fields you get extra terms with your geodesic equation and the solution won't be a geodesic anymore.

I'm not sure if I understand your question.

 

[pervect]

If the OP is just looking for some basic information about the principle of least action, E F Taylor's website might help.

http://www.eftaylor.com/leastaction.html

There are a LOT of papers about the topic there - it's a bit of a crapshoot to guess which must be most helpful, but I'd say http://www.eftaylor.com/pub/ForceEnergyPredictMotion.pdf is one of the more elementary ones, as it manages to avoid using Lagrangian mechanics. I suppose I should mention that their approach is a bit unusual, but that's part of the appeal - most "standard" appraoches require a lot more knowledge.

Some might not like the "let's take a guess and then show that it gives the right answer" approach, though...

 

[runner108]

Thanks Pervect, I'll look into it.