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In mechanics problems, especially one-dimensional ones, we talk about how a particle goes in a direction to minimize potential energy. This is easy to see when we use cartesian coordinates: For example, $-\frac{dU}{dx}=F$ (or in the multidimensional case, the gradient), so the force will go in the direction of minimizing the potential energy.

However, it becomes less clear in other cases. For example, I read a problem that involved a ball attached to a pivot, so it could only rotate. It was then claimed that the ball would rotate towards minimal potential energy, however $-\frac{dU}{d\theta} \neq F$! I think in this case it might be equal to torque, which would make their reasoning correct, but it seems like regardless of the degrees of freedom of the problem, it is always assumed that the forces act in a way such that the potential energy is minimized. Could someone give a good explanation for why this is?

Edit: I should note that I typed this in google and found this page. where it states that minimizing potential energy and increasing heat increases entropy. For one, this isn't really an explanation because it doesn't state why it increases entropy. Also, if possible, I would like an explanation that doesn't involve entropy. But if it is impossible to make a rigorous argument that doesn't involve entropy then using entropy is fine.

As a side note, how does this relate to Hamilton's Principle?

Kyle Oman
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user35734
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7 Answers7

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This is a physical rather than a mathematical justification - ignore my answer if that isn't what you wanted!

All systems have some thermal motion so they explore the phase space in their immediate vicinity. If there is a nearby point with a free energy lower by some amount $\Delta G$ then the relative probability of finding the system at that point will be $\exp(-\Delta G/RT)$. So if the energy is minimised by moving to that point, i.e.$\Delta G < 0$, we just have to wait and we'll find the system has moved there. The only place in phase space the system won't move is when the free energy is at a (local) minimum. That's why a system always (locally) minimises its free energy if you wait long enough.

John Rennie
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Note the title of the link you give :

Minimum total potential energy principle

bold mine.

The only answer to "why" questions about principles in physics is "because the theoretical models dependent on it have been found to describe all known data and can predict new ones".

Why questions in physics when they hit postulates and laws, is like asking why for an axiom in mathematics. Principles are part of the definition of the theory, and physics theories are established when validated continuously by the data.

One cannot explain this principle except by attributing it to observations that forced us to use it axiomatically.

anna v
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The question about minimizing potential energy and the replies that such questions do not make much sense is a typical conversation between a physicist and a mathematician:

Physicist: - Why systems tend to minimize potential energy?

Mathematician: - Look around, lots of things follow this principle: potential energy, entropy...

Physicist: - OK, I can see that, and that's why I'm asking. Why? Why lower number is better? What's so special about that? Why not bigger numbers?

Mathematician: - Don't ask such questions. This was proved by experiments. We just must accept it as a fact, as an axiom. And axioms just are. So don't get metaphysical.

... and this is where the poor inquisitive physicist quits asking any further if he wants to pursue his career, as the label of a "philosopher" might not be very helpful ...

Well ... the mystery of the tendency to minimize potential energy evaporates quickly as soon as you ask yourself other questions: What this potential energy comes from? What it represents? What is the underlying source/cause?

In the case of the ball rolling down the slope the answer is obvious. This cause is gravity. Massive bodies simply attract all other massive bodies around, and since gravitational potential is defined in terms of distance to the source (centre) of gravitation, then obviously the ball that is moving toward the centre of gravity is minimizing its potential. So the important question here is: what is more fundamental - gravity or gravitational potential? Is the axiom of minimizing potential some ultimate answer? Or ultimate cause? In case of gravity, the potential just tells you in a way how distant the attracting bodies are, and therefore whether the force can produce any movement (and how much of this movement) or not. Minimum potential means that no movement due to the force is possible; maximum potential means that (a lot of) movement is possible. And the source of movement is the force, not the potential. And the movement caused by the force can only go from larger numbers (distance) to smaller numbers. That's how it works: the force pulls things in, so the distance becomes smaller. And that's how the potential has actually been defined. Therefore the answer is not that Nature (for some metaphysical reasons) developed a liking for small numbers. It is rather that Nature provides this very fundamental attractive force - called gravitation - which makes matter "come together". And the tendency to lower the potential is a way of saying that the attracting force causes bodies to come together, and thus lower the distance between them.

To sum up our gravity case: The tendency of a system to minimize potential energy results from underlying force pulling a body toward its source. (or pushing a body away, if the nature of the force is repulsive). This results in minimizing distance, therefore in minimizing potential energy. So more potential energy equals more potential movement, and less potential energy equals less potential movement. (Obviously, if the two attracting bodies are already together, the force is not able to produce any movement, hence zero potential energy.)

Now, this reasoning is equally true for electric potential energy or elastic potential energy. In all of these cases there is an underlying force that makes matter move. So if the nature of the force is attractive, the related potential energy grows with distance, which represents potential movement these forces can produce.

So you were perfectly right to ask "why" about this axiom. This way you are more likely to produce more profound answers that will lead you to more fundamental phenomena, and ultimately to better understanding Nature. There is probably a limit to understanding why things are the way they are, but the tendency to minimize potential energy is certainly not the Ultimate Cause.

By the way, you said you would rather not get the answer in terms of entropy. You were right, because entropy is not really necessary to find the answer about the potential. Yet the case with the ball shows us a very interesting thing - gravity and entropy are two basic opposing phenomena. Gravity counteracts entropy. Gravity pulls molecules together, while entropy makes them move away. It's easier to understand this after the term "entropy" is clarified and demystified. Entropy is simply the phenomenon associated with the movement of molecules (or particles). Molecules with higher energy, i.e. moving faster tend to travel to regions where there are more molecules with lower energy, i.e. moving slower. Why? Sure you can ask why, this is also no ultimate axiom or metaphysics. This is simply because molecules moving faster are more likely to bump each other when they are grouped together. But they are less likely to bump those moving slower. The consequence is that fast moving molecules are restricting each others movements, as they hit each other more often. Therefore, the fast moving molecules will more often penetrate areas where there are a lot of slower moving particles. In any given time period the slower ones "occupy" less space thus leaving more room for the fast ones to move (in a given $\Delta t$ necessary for a fast molecule to travel $\Delta x$ that equals its size, a slower particle will travel less distance than a faster one). And when the faster molecules mix with the slower ones, they will give some of their speed (energy) to those moving slower (through collisions, which also do occur although less often). This way the system goes from "higher entropy" to "lower entropy". But this happens not because Nature just likes small numbers better, as you can see. It can be perfectly explained mechanically without assuming axioms.

bright magus
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As anna v said, principles are like mathematical axioms. A principle is correct if the laws of physics that can be derived from it correctly describe all experimental data. Some people choose the Newton's laws as the basis of mechanics. Some people choose principle of least action (AKA Hamilton's principle).

You can derive the principle of least action from Newton's laws. These Lectures on Classical Dynamics do it in a few condensed pages in chapter 2.

The principle of least action describes the motion of a particle in terms of paths. A path starts at one position and momentum, and ends at another. The path describes the position of the particle as a function of time. This means the path also describes the velocity and momentum of the particle.

Many paths connect the start and end. The principle of least action tells you how to find the one path for which Newton's Laws hold at every point.

The principle of least action can be generalized to cover all the laws of physics.

You may find chapter 1 of the Lectures relevant. It talks about energy and derives conservation laws for single particle and many particle systems.

A particle doesn't always go in a direction that minimizes potential energy. For a mass and spring, the potential energy goes up and down. For a planet in a circular orbit, it stays constant.

It is true that force points in the direction that minimizes potential energy. This is follows immediately from the definition of potential energy.

The forces on a particle are often purely a function of the particle's position, not of the particles velocity. In this case, there is another way to describe the force. You can define a quantity $U$ that depends only on position, where $U = - \bigtriangledown F$. $U$ is the potential energy.

The reason for defining $U$ is that it is useful. You can show $U + T$ is conserved. Given $T$ at one position, you can use the conservation law to calculate $T$ at any other position without having to know all the forces in between.

Entropy is used when you have so many particles that you can't keep track of them all. You have to fall back on properties that summarize the motion.

For example, temperature summarizes the average kinetic energy in internal vibrations of the particles that make up a rigid body.

mmesser314
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We need to seperate two things here:

  • Force does not minimise potential energy. Rather you could say, "it tweaks the momentum in such a way that the potential energy can not rise indefinitely".
    Incidentally, the way this works (2nd-order differential equation → oscillatory solutions) tends to bring the system always back to a state of low potential energy, but this isn't really a necessity (think about the solar system, where the potential energy of the planets stays more or less constant).

  • The minimum potential energy principle is quite another issue. It assumes there are a great lot of microscopic degrees of freedom (molecular rotation & vibration, electron waves etc); such are always necessary for frictional forces to work.
    Very simply put, this law is just the equipartition theorem: if you put some energy in the system's macroscopic degrees of freedom, that energy will over time "leak" into the microscopic degrees until each degree has on average the same (very small) energy. Because there are so much more microscopic than macroscopic degrees, but they are "invisible", it will seem as if the potential energy decreases.

leftaroundabout
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This used to be unintuitive for me as well. First, recognize the general definition of work.

$W = \int \vec{F_{net}} \cdot \vec{ds}$

Now, taking the gradient on both sides:

$\nabla W = \nabla \int \vec{F_{net}} \cdot \vec{ds}=$

$\int \nabla\vec{F_{net}} \cdot \vec{ds}= \int \sum_{i}\, \frac{\partial \, \vec{F_{net}}}{\partial\, x_i} dx_i$

where $x_{i}$ indicates a position coordinate. Continuing:

$\int \sum_{i}\, \frac{\partial \, \vec{F_{net}}}{\partial\, x_i} dx_i= \int d \vec{F_{net}} = \vec{F_{net}}$ So: $\nabla W =\vec{F_{net}}$

Note that the gradient operator $\nabla$ gives a vector of the maximum change of some function with respect to its independent variables, in this case, the position coordinates. So, all that the previous equation says is simply that the maximum increase in work will be in the direction of the net force on the system (the line of action). This seems intuitive, and is actually the more general statement of what I’m about to show. Now, for a conservative system- a system with only conservative forces acting on it- we define the potential energy function from the work function by:

$U = - W = - \int \vec{F_{net}} \cdot \vec{ds}$

So, given the formula derived before:

$\nabla W =\vec{F_{net}} = -\nabla U$

This is the famous formula for conservative systems that connects the force and potential energy. It says something similar to the work. The greatest spatial decrease in the potential energy is in the same direction as the net force on the system. Because the system’s motion will be governed by the net, nonzero force on the system, which is in the same direction as the greatest spatial decrease as the potential energy, a conservative system will follow the path that minimizes its potential energy. In sum, all that one is saying when they say a conservative system wants to minimize its potential energy is that a system will follow the direction of the net, nonzero force on the system.

I hope this helped!

Brian Opatosky
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Its just chance,most of the universes without this function are semi-chaotic and shortlived. Its like saying why use eggs in omlettes ,'cos its not omlette without eggs. As to WHY , well there are many rules(laws) to make a universe where life exist ,that is its purpose. love

user46796
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