Are There Extremal Action Paths in Physical Systems Beyond Minimal Action?

[jostpuur]

Is there any physical systems, where there would be paths of extremal action, that are not paths of minimal action? I'm having doubts on this, because the oscillation argument that is used in derivation of the Hamilton's principle out of Feynman's path integrals, seems to imply precisly minimum of action, and never maximum.

 

[CompuChip]

I believe the entire point is, that nature always tries to do things at minimum cost. That usually means taking the path of minimal action. Even in cases where an energy-consuming operation is performed (for example, splitting a liquid to a gas) this is because it will yield more energy afterwards (in this case, a gas would have more degrees of freedom -- hence a much higher entropy).

 

[StatusX]

I'm having doubts on this, because the oscillation argument that is used in derivation of the Hamilton's principle out of Feynman's path integrals, seems to imply precisly minimum of action, and never maximum.

Actually Feynman's path integral approach only necessitates stationary action paths, not minimal or maximal ones. The idea is that a "sum" is taken over all paths of something proportional to $e^{iS/\hbar}$. In general, this oscillates so rapidly as you move from path to path (at least in the classical limit, due to the $1/\hbar$ factor) that the integral is very small. It is only when the action is stationary at a path, so that the function is approximately constant on a neighborhood of the path, that there is a non-zero contribution to the sum from the path, and so such paths are the physically relevant ones.

 

[jostpuur]

Actually Feynman's path integral approach only necessitates stationary action paths, not minimal or maximal ones. The idea is that a "sum" is taken over all paths of something proportional to $e^{iS/\hbar}$. In general, this oscillates so rapidly as you move from path to path (at least in the classical limit, due to the $1/\hbar$ factor) that the integral is very small. It is only when the action is stationary at a path, so that the function is approximately constant on a neighborhood of the path, that there is a non-zero contribution to the sum from the path, and so such paths are the physically relevant ones.

Perhaps I made a mistake on this, but I'll still ask about an example of such system where maximum action paths exist.

 

[smallphi]

The action functional in General Relativity is the proper time along the curve. In flat Minkowski spacetime, bodies in free-fall maximize the proper time along their path. An example of that is the twin paradox - the twin staying on Earth is in free-fall and follows the curve of maximum proper time, hence ages much more than the twin that travels to the star and back.

A more down to Earth example is to consider a sphere and to ask which are the curves that extremize the distance (in the role of action) between two arbitrary points on the sphere. If the two points are not antipodes, there are only two such curves, parts of the great circle passing through the two points. One of the curves minimizes the distance/action, the other maximizes it.

 

[StatusX]

Following what smallphi said, here's an example from general relativity. Let one particle move in a circular orbit around a planet with period T. Then another particle initially at the same point as the first is launched directly away from the planet with a velocity below escape velocity such that it returns to the starting point in time T. Then both particles have taken different paths through spacetime between the same two points. I'd have to do the calculation explicitly, but I'm betting they have different amounts of proper time. In any case, it shows that there's more to determining the path between two points than extremizing the action.

 

[smallphi]

The example given by StatusX shows that there are several free-fall orbits connecting the same two points in the spacetime around a planet. All these orbits extremize the GR action/proper time between the fixed initial and final points. It's not clear if they all maximize it, minimize it or some maximize some minimize.

By 'maximize' or 'minimize' we mean local maximum/minimum not absolute. That is why it is possible to have several curves extremizing between the same two points.

 

[pervect]

To re-use and old post on this point in a thread that went sour:

From: http://www.eftaylor.com/pub/call_action.html

The action for an actual path may be a saddle point rather than a minimum. So the most general, but rather esoteric, term for our basic law is the principle of stationary action.

I don't, however, have an actual specific example of this arising in a physical system.

 

[jostpuur]

I don't know general relativity well yet, so I cannot say anything about extremizing proper time.

Your sphere example however doesn't seem fully correct. If a particle is forced to move otherwise freely, but on some sphere surface, then there usually is two straight paths between two given points. Even though other one is longer and other one shorter, they are both local minimums of the action, aren't they?

 

[smallphi]

The bigger part of great circle between the two points seems to be a saddle point of the distance integral between the two points. It is shorter than some nearby paths (ones that oscillate around the great circle) and longer than others (ones that always remain on one side of the great circle).

 

[olgranpappy]

Perhaps I made a mistake on this, but I'll still ask about an example of such system where maximum action paths exist.

I can give you a silly example. Take the action $$S$$ of any physical system with which you are familiar and which is minimized by the classical path... and then make the replacement
$$S \to S'=-S$$. The new action $$S'$$ is maximized by the classical path.

 

[smallphi]

Good one :) but then we could fix the overall sign of the action by choosing the usual sign convention that the Lagrangian is always Kinetic-Potential energy not the negative of that. Then we could ask the question if that action is minimized or maximized.

 

[olgranpappy]

Here's another slightly relevant example. Consider the method of "steepest-descent" from complex analysis. This is exactly the same as a "maximum action" principle rather than a "minimum action" principle (which would be the method of "steepest ascent")... the only difference being that in complex-analysis we usually care about the maximum of a function at a particular point in the complex plane rather than the maximum of a functional at a particular path. Again this could be turned into a minimum principle by seperating out a factor of -1.

 

[olgranpappy]

Oh, I finally found a "standard" maximization principle: In section 10 of Landau vol. 8 there is defined a standard "Free Energy for constant potentials" $$\tilde F$$. This is minimum with respect to most of it's variable but later on in section 18 we see that a maximization principle rather than minimization with respect to the fields of $$\tilde F$$ yields the field equations.

 

[olgranpappy]

...of course, it's still just a sign convention.