[paros (OP)]

Which one of these sentences best depicts the Principle of Least Action? And are any of these completely false?

• A system that is under stress or strain will evolve in such a way as to relieve that stress.
• Physical systems attempt to convert potential energy into kinetic energy as fast as possible.
• The total energy of a system is conserved.
• Physical systems tend to maximize potential energy. (?)
• A physical system evolves in such a way as to keep the the kinetic energy nearly equal to the potential energy, if it can.

Before answering, I'm searching for an underlying philosophical principle, not more equations and more jargon. We all know what the equations say, and we can all go to wikipedia and find out how to define "action" as an integral and then demand the change in the path is a 'stationary point' in terms of calculus of variations. Yes and yes. We all know this already. We can bog each other down in jargon until we are blue in the face. But I'm trying to pin down what the equations MEAN in terms of the ontology and dynamics of the physical world.

[imatfaal]

1.  stress and strain are terms of art in physics; and within those definitions this part is simply not correct.  If we do not use the accepted terms then 1 can mean almost anything.

2. but they do not.  the conversion of U to K is dependent on the mechanism - and will happen maximally only to the extent of the mechanism and other constraints.  everything that is not forbidden is compulsory - but lots is forbidden or constrained.
 
3. is dodgy for this definition - whilst completely true.  The principle of least action is that the integral of the Lagrangian is minimized - but, as this is intimately connected with total energy in most investigations. Energy is conserved this is required for the PofLA - but Emmy Noether's work (which shows conservation laws through symmetries) relies on the the PofLA for its proof.  It is too cyclical (although all definitions must be eventually)

Surely 4 is just plain wrong - an apple in mid air has a higher (ie closer to zero from the negative side) U than an apple on the ground.  Nature seeks energy situations which are stable which normally means that there is no easily exploited potential to move them to other situations.  It is the route - not the endpoint - that is crucial in the PofLA; in the route you are looking for minimizations of Kinetic less Potential

5.  I don't think this is correct. 

"Physical processes take the easiest route that their constraints allow"

[alancalverd]

There is no underlying philosophical principle. Philosophy is a human construct, not a natural phenomenon.

3 is true of a classical, closed, isolated, adiabatic system. The other statements are nonsense.

Evolution is an emergent property of active biological systems, which are themselves a small subset of chemical systems, which are a particularisation of physics at the molecular level. The word cannot be generalised to the whole of physics. 

[paros (OP)]

alancalverd,

This is a miscommunication. I meant to use the phrase "evolve in time", and I was not referring to biology.

[paros (OP)]

The principle of least action is that the integral of the Lagrangian is minimized - but, as this is intimately connected with total energy in most investigations.

Exactly. But what does this mean ??

Could you expand on what you mean by "intimately connected" ?

To answer this question, can you describe a universe (with conserved energy) in which least action is not abided by?  What sorts of differences would we see with falling apples and trajectories of objects and such?  (...or use any other toy example that best describes these differences)

Now those questions can be answered in several different ways. One way to answer them would be saying,  "If energy is conserved, it must be the case that the integral over the Lagrangian is minimized".  i.e.  PoLA is a different way of phrasing energy conservation. If one, then the other.  If energy conserved, then PoLA falls out at the bottom of the proverbial chalkboard. 

Is that true?  Or rather is PoLa a separable, disconnected property added on top or besides energy conservation?  Or are they two ways of stating the same thing? 

Again, the best way to answer these questions is describe some toy scenario in which energy is conserved, but PoLA is not abided by.

My best understanding at this point (today) is that the larger potential energy becomes, the faster it is converted back into kinetic energy, and the lower it becomes the slower it is converted.  I'm looking for some sort of rule-of-thumb way of understanding PoLA as well as  Hamilton's principle.  I know exactly what the equations say and I can repeat them back to you on a test. I can blindly apply Hamilton's principles to exercises in a textbook with pinpoint accuracy and get my A+ in the class.   But I want something more.  I want a more well-rounded understanding of what this means for our universe.

[Pmb]

But I'm trying to pin down what the equations MEAN in terms of the ontology and dynamics of the physical world.

There are two different but related principles. One is the Principle of Stationary Action, aka Hamilton's Principle also known as the Principle of Stationary Action, while the other is the Principle of Least Action. The two are often confused.

See
http://en.wikipedia.org/wiki/Hamilton's_principle
http://en.wikipedia.org/wiki/Principle_of_least_action

The Principle of Least Action states that the integral of the functional is such that the action is a minimum, i.e. happens in the least possible amount of time (whatever that means).

Otherwise it says exactly what it means. If there was another way to express it in non-mathematical terms then you'd probably already know it. The fact that it can't is the reason it's stated as such. That's why its a known fact that to properly understand physics you must know the language of physics, i.e. math.

To illustrate my point let us consider the more simple definitions that you find in physics such velovity, force, the curl of a vector field, etc.

We all know what velocity means. What about momentum? Do you know what it means? What about Energy or the stress-energy-momentum tensor is and what it means?

Action is defined in terms of what is called a functional. In Hamilton's Principle the functional is the Lagrangian. In the Principle of Least Action it's something else (see link above). The Lagrangian is defined as the difference between kinetic and potential energy. The action is a time integral and is a line integral evaluated along a path in configuration space.

I wish I could be of more help but that's the nature of the beast and a good reason why we need to know math to know physics.

[alancalverd]

alancalverd,

This is a miscommunication. I meant to use the phrase "evolve in time", and I was not referring to biology.

Nevertheless, the sentence is drivel. Consider a falling apple. All the potential energy is converted into kinetic energy, initially of the apple and eventually of the surrounding earth and air. 

We all know what velocity means.

Alas, 'tis not so. I suspect that at least 90% of the population (and 100% of journalists or politicians) think it is the same as speed, despite 11 or 12 years of compulsory schooling. 

[Pmb]

Which one of these sentences best depicts the Principle of Least Action? And are any of these completely false?

In addition to my post above I should/could say the following:

A system that is under stress or strain will evolve in such a way as to relieve that stress.

False. A particle on the end of a perfect spring will undergo harmonic motion. The stress in the spring with oscillate, not relieve the stress.

Physical systems attempt to convert potential energy into kinetic energy as fast as possible.

False. Each system will evolve according to the parameters of the system, not as fast as possible.

The total energy of a system is conserved.

False. The total energy of the system is not always conserved. E.g. if the Lagrangian is a function of time then the energy will not be conserved.

Physical systems tend to maximize potential energy.

False. The maximum energy is set by the total energy of the system and is not a maximum. At best a physical system will tend to be minimized but such systems are not represent able by a Lagrangian and therefore The Principle of Least Action will not apply

A physical system evolves in such a way as to keep the the kinetic energy nearly equal to the potential energy, if it can.

False. E.g. in a period system the potential energy will only be equal to the kinetic energy twice in its cycle.