# Principle of minimum potential energy -- why is it true?

Dear all,

Two fundamental concepts from thermodynamics are that systems try to minimise their potential energy, whereas the tendency is to maximise the entropy (ofcourse I am not at all precise in the above statement, but I hope I am able to convey the matter.  Further, the two can be combined in the form of a free enrgy as applicable to the system under consideration).

In case of entropy, the statistical mechanics explains by saying that the number of microstates corresponding to a disordered arrangement is much more than the number of microstates corresponding to an ordered arrangement.  If we assign equal probabilities to all microstates, this would mean that there is a higher probability for states corresponding to higher entropy.  Hence, isolated systems tend to increase their entropy over time.  (Please correct me, if I am making any fundamental mistakes here.)

My question is: whether there is any such explanation for principle of minimum potential energy?  In other words, why a system should try to minimise its potential energy? Is it just an observation or hypothesis?

Thanks for sharing your thoughts on the matter...

Regards,

### the minium potential energy

the minium potential energy is a hypothesis that has been confirmed by macroscopic observation - thus becomes a theory. Why things like to minimize the potential energy? whatever answer it is we can have continuous why going on the line. But I think one contribution of that is to make the universe stable - a fundamenal condtion to produce intelligence that such why questions can be asked

In thermodynamics I think such a minimum principle becomes a probabilistic concept. If some people do observe something different, like a man walking on the water,a supposed extreme impossibility becomes a fact, then a different theory has to be proposed to improve the previous one at least in that particular situation.

### Re: the minium potential energy

Thanks for your comment...  Can you please elaborate on how the tendency of systems towards minimum potential energy contributes to make the universe stable?

### Hi Jayadeep,  This is a

This is a tricky question. "the univerise is stable" is subjectively said so as statu quo - seemingly entering into the rough landscape of tautology, thermodynamic stability is like another way of saying minimum potential energy......you see the continous why is going on

In  a big picture I think all human's scientific knowledge, similar to religious proposals , is subjected to questioning of tautology. We propose some hypothese, then use our empirical observation/experience to validate or falsify them. However our intepretation/understanding of empirical observation/experience itself is subjective, and is fundamentally influnced by the teachings (hypothese)we accept.

### Re: Minimum potential energy

That's a good question. Let me jot down my on-the-fly thoughts. Not very well organized, but let me give it a try, anyway.

1. What context do you have in mind? Elasticity? General thermodynamics? Somewhat analogous situations such as those from optics? ... Let me presume elasticity here.

In elasticity, it's the total potential energy [^]; you have to add (with the correct algebraic sign) the work done by the applied forces as well.

2. Also, in the strict mathematical sense, the procedure of setting the second derivative to zero implies a point of stationarity rather than that of a minimum

3. The Wiki page says that it's just the principle of maximization of entropy, as applied in (or specialized to) the context of elasticity. A similar viewpoint is echoed at many places.

However, I have no idea why Prof. Sanjay Govindjee might have said in his notes that it has nothing to do with conservation of energy (emphasis his) [^]. Perhaps he could provide some comments?

4. Stationarity by itself implies that the point in question could be a maximum, not necessarily a minimum. I can't off-hand think of an example from elasticity where a maximum applies---perhaps an example could come from structural instability? In optics, for Fermat's principle (another stationarity principle which is applicable in that context), it's possible to cite an example: the case of the light reflection from a convex surface follows the maximum, not the minimum.

Also, there are issues like the global vs. local minima; however, these could be ignored for the issue of the maximum vs. the minimum.

5. A Google search throws up this paper by Wen and Zheng [^] early, and it seems relevant here, esp. the first section. A few notes: De Alembert's principle simply makes dynamics accessible via the usual equilibrium relations. The principle of virtual work applies regardless of the nature of the constitutive law, and rests on de Alembert's principle for its validation. The principle of total potential energy is simply an application of the principle of virtual work.

6. Thus, the physical content of this principle is the same as that of Newton's laws of motion, though it falls in that line of development---broadly speaking, the energetics program---which was initiated by Leibnitz. The only difference between the two approaches (the Newtonian approach based on vectors (momentum conservation) and the Leibnitzian approach based on scalars (energy conservation)) is one of mathematics, not of basic physics. The energy methods are mathematically more convenient in finding solutions using the analytical techniques of mathematics in those cases in which boundary conditions are too complicated to handle using the method of momentum conservation, e.g., an infinite number of collision boundary conditions as in the simple case of a motion of a body restricted by a constraining surface (think of a round steel ball moving through a tube or a bead moving on a curved wire). Or, in solid mechanics, the statically indeterminate cases.

Inasmuch as Newton's laws are inductive generalizations, and these principles are equivalent to Newton's laws, it's obvious that these principles, too are inductive generalizations. Thermodynamics further generalizes, in a way, these originally mechanical principles when it comes to the context of studies of heat, mechanical work, their relationships, and effects.

Qua inductive generalization, these principles have no prior deductive proof. Their truth is established using the method of induction. However, qua their use as starting points in performing analysis, their status may be taken as that of postulates.

I hope this discussion addresses that part of your question which deals with whether it's an observation or a hypothesis. Answer: both, and neither! It is an observation, but it's not just an isolated instance of a concrete observation; it's far more than that; it's a principle, an inductive generalization. And qua a validated inductive principle, it's not a mere hypothesis---given its inductive context, it's a statement of truth, entirely valid within that context (including its applications).

7. You wondered about a statistical mechanical view or interpretation of this principle. In the context of solid mechanics, I don't think any one has worked it out.  Working it out would be a good research topic. I have put forth a conjecture during my PhD studies that it should be possible to model tensor fields of elasticity using the techniques like the random walk. It still is a conjecture .

Hope this helps. Over to iMechanicians for any corrections or expansions.

--Ajit

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[E&OE]

### Re: Minimum potential energy

Dear Ajit,

Thank you for the detailed comment.  I need to spend more time on the reply as well as the documents you referred to.  Probably I will have more questions after that...

Meanwhile you mentioned a connection between principle of minimum potential energy and maximization of entropy.  That appears to be a very interesting concept.  Can you please elaborate?

### Re: Minimum potential energy

I guess the abovementioned Wiki page is clear enough, isn't it? You will find the same argument elsewhere too. It's enough to locate the elastic strain energy as being a part of the internal energy in the thermodynamic scheme, and the rest is straight-forward. At least, it should be. Unless, of course, you have something else in mind.

Nevertheless, to elaborate, just because you explicitly ask for it: See the Wiki page on the Internal energy [^]. Refer to the section: "Internal energy changes." Ignore $Q$ for a typical beginning elasticity analysis, and the $W_{\text{extra}}$ term is where the elastic strain energy would reside. Ignore gravity, EM, etc., and you may presume the $W_{\text{extra}}$ term to solely consist of the elastic strain energy. Now, scroll further down to the section: "Internal energy in an elastic medium," and you have an expression involving all: $U$, $S$, and the elastic strain energy (whether the constitutive law is linear or nonlinear). It's an equation being presented there on the authority of Landau and Lifshitz! What else could one possibly ask for? ...

... What else? ... Oh well, may be, Truedell? [^]. ... May be. I haven't read it. In fact, not even Landau and Lifshitz. Not L&L v. 7, because I browsed through their v. 3 on QM and immediately concluded that their treatment lied between unsatisfactory to outright poor, and for my purposes, elasticity as presented in the usual texts was enough (the thickest being Love). As to T: I didn't even flip through it more than once (some two decades ago) because it's an information-wise dense tome (or so I vaguely recall as being my impression back then), whereas my interests have been elsewhere. As such, I don't have any particular opinion on it. But guess T would present essentially the same viewpont if not exactly the same equation.

So, there. I guess you may get more detailed treatments from L&L and/or T. Anyway, did you have some other viewpoint on these matters so that you would seek an elaboration?

Ajit

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[E&OE]

### Total Potential Energy as a Variational Principle

Hi Friends,

Total Potential Energy is a Classical Variational Principle which use the Formulation of Strain Energy and External Works.

Strain Energy is obtained from Internal Forces generating Stresses. This Formulation is Trivial in Linear Elasticity as you can Understand

the theory by the example of a Loaded Spring. But this Formulation can't be Applied to All Applications because of other

Complex cases like Considering the Stresses as Unknowns. Thus you can Classify your Application in another Variational Principle as :

Hellinger-Reissner, Complementary Total Potential Energy, ..

Sincerely,

M.L.

### becasue of the way we define the potential energy

It is an interesting question. In my opinion, it is true becasue of the way we define the potential energy.

The potential energy is a sort of "potential ability of generating work output". We believe that a system must change its thermodynamic state so that it can generate a (net) work output, and the system losses some of its "potential energy" due the change of state. A system can change its thermodynamic state due to external or internal stimuli.

We say that a system is in an equilibrium state, if its thermodynamic state does not change unless an external stimulus is applied. We also believe that a system in equilibrium does not do work to the environment. So we draw the conclusion that in equilibrium, the internal stimuli cannot further decrease the potential energy of the system. In other words, the potential energy is minimized at equilibrium states.

Notice that I use two "believe"s above. Those are the hypotheses of a system to which the principle of minimum potential energy applies. If, in contrast, a system CAN generate a net work output without changing its thermodynamic state, or the system CAN automatically do work to the environment even in equilibirum, the principle fails.

Systems compatible with the principle of minimum potential energy form only a subset of thermodynamically accessible systems. You can call such systems conservative systes, or, within the content of elasticity, hyperelasticity.

p.s. all I said has nothing to do with D'Alembert's principle in dynamics, which is another story.

Best,

Yintao

### Minimizing free energy

I have tried to discuss the minimization of free energy in my notes on free energy.  Hope this discussion helps.

### Minimization of Total Potential Energy

Check the Solution of your Equilibrium State of the Functional (Function of Functions) with dEp=0 after Differentiating the Total Potential Energy Ep. If it is a Linear System of Equations with a Symetric Positive Definite Matrix this corresponds obviously to a Minimization case.

Mohammed Lamine

### thank you jayadeep i more

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