# Minimum Potential Energy

## Summary:

Catenary Cable Shape
Many, many years ago while in engineering graduate school I was studying calculus of variations. One classic problem was to determine the shape of a hanging cable supported at its two ends. After minimizing the integral, the catenary curve was the solution. The basic assumption in setting up of the integral to be minimized was that the potential energy of the cable must be a minimum. Why must the potential energy be a minimum?

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anorlunda
Staff Emeritus
Why must the potential energy be a minimum?
Because physical systems naturally find the minimum. For example, the ball rolls down the hill to the valley.

Mathematcially, it seeks a stationary point. Hill tops and valley bottoms are both stationary, but a ball at the top of a hill is unstable; push it in any direction and it rolls down. A ball at the bottom of a valley is stable; push it in any direction and it returns to the bottom.

But why do physical systems seek the minimum? The ball near the bottom of a valley finds its resting place at the bottom which, coincidentally, is the point of minimum PE.

PeroK
Homework Helper
Gold Member
Summary:: Catenary Cable Shape

Many, many years ago while in engineering graduate school I was studying calculus of variations. One classic problem was to determine the shape of a hanging cable supported at its two ends. After minimizing the integral, the catenary curve was the solution. The basic assumption in setting up of the integral to be minimized was that the potential energy of the cable must be a minimum. Why must the potential energy be a minimum?
That particular problem can also be solved by looking at the tension in the wire. This turns out to be equivalent to minimising the potential energy.

In general there are two formulations of classical mechanics: a direct use of Newton's laws; and the use of the Lagrangian principle, which involves minimising a certain quantity associated with a system. For dynamic problems, these can be shown to be equivalent.

The Lagrangian principle extends widely, especially into areas where there are no forces - like GR (General Relativity) and QM (Quantum Mechanics), where a similar reformulation due to Hamilton is used.

Another example is that when light refracts at a boundary it does so in accordance with Snell's law. On the one hand this is a solution based on what happens solely at the boundary. But, the interesting thing is that resulting trajectory effectively minimises the time that light takes to travel between any two points.

In general, therefore, the Lagrangian principle is deep rooted in nature. And Newton's laws can be seen as one manifestation of this, suitably reformulated.

You could take a look here to start:

https://en.wikipedia.org/wiki/Lagrangian_mechanics

• tech99
"But why do physical systems seek the minimum? "
'Seek' may not be the right word. Using the ball/hill example:

First: Absent friction, the ball will just endlessly roll back and forth across the valley

At any time, the ball has a total energy which is the sum of potential and kinetic energies. for the 'absent friction' case, the total stays the same, but the individual quantities change by equal/opposite amounts (like a slinky, if you're old enough to know what that is). PE is 0/minimum at the bottom of the valley; KE is 0 at the high point on each hill.

Friction acts to dissipate kinetic energy. With friction, there is a little less total energy on every cycle of the ball; the ball 'stops' a bit lower on hill each cycle because there was less KE to 'trade' for PE. The ultimate result is the dissipation of all of the KE; the ball stops at the low point of the valley with nothing to trade for an increase in elevation.

• gmax137
Why must the potential energy be a minimum?
You can set it to the other extreme too and it'll still give you a valid answer (for the case of negative gravity - ooops )

Basically, you are seeking stable solutions, with the sum of energy of the whole system being constant.
With losses eating up (removing) any excess/kinetic energy, 'stable' constant would mean 'minimal' on long term.

Ebi
But why do physical systems seek the minimum? The ball near the bottom of a valley finds its resting place at the bottom which, coincidentally, is the point of minimum PE.
Every system naturally finds its minimum energy because minimum energy means maximum stability.