# How come there is no conservation of force?

1. May 9, 2010

### polaris12

title says it all

2. May 9, 2010

### Staff: Mentor

The title is not very well worded, but if you mean, 'why doesn't every force have an equal and opposite force associated with it?' then the answer is, they do.

3. May 9, 2010

### polaris12

I'm sorry, but what I meant to ask was that I learned about how momentum and energy are always conserved, but my teacher never mentioned the same thing for force. Was that just an oversight or does it really not exist, and if so, why not?

4. May 9, 2010

5. May 9, 2010

### Phrak

I hadn't really thought about it. Excuse me if the mathematics is too advanced for you. It's a very thoughtful question.

Where conservation of momentum over a region might be expressed as

$$\sum \frac{d\textbf{p}}{dt} = 0 \mbox{~or~} \sum \textbf{F} = 0$$

then the conservation of force is already implicit in the conservation of momentum.

$$\sum \frac{d^2\textbf{p} }{dt^2} = 0$$

We could go on and on with this and say that impulse, and so forth, are conserved.

6. May 9, 2010

### Staff: Mentor

I don't really like that - I don't think it addresses the OP's question and I'm not really sure it is correct anyway. The definition of "conserve" is "to hold (a property) constant during an interaction or process". What I tried to convey is that this word doesn't really apply to force, which the OP already instinctively knows, which is why the question was asked.

Consider a common conservation of momentum problem: with a pair of billiards balls. The "interaction or process" is a collision. The total momentum before, during and after the collision is always the same. But the force starts off at zero, increases, then decreases back to zero. So too with impulse.

7. May 10, 2010

### Phrak

In common usage a conservation law is something invariant over time.

Consider your collision of two billiard balls. During every collision dp/dt of one billiard ball is equal and opposite to the dp/dt of the other. That is to say, the sum of the force vectors is conserved.

8. May 10, 2010

### Mu naught

I think the idea of "a force" is really where the confusion lies. Force I think is really an unnecessary, but convenient concept we use to make abstractions about the world. If we think in more fundamental terms like work, energy and potential then I think we eliminate all these confusing ideas about "forces".

9. May 10, 2010

### Andy Resnick

I'm with Phrak- this is a very thoughtful question. Here's how I would answer:

Unlike energy and momentum which are properties inherent to the object (or system), a force represents an interaction of the system with something else- gravity, electromagnetism, friction, etc., all represent two objects interacting with each other.

Energy and momentum (as well as mass, charge, and some other properties) obey 'balance laws', which are very general expressions relating the total change (of a quantity) to the internal changes plus any amounts flowing into or out of the system:

http://en.wikipedia.org/wiki/Continuum_mechanics#Balance_Laws

'Force' does obey a balance law because it is not a property of a single system- it is an interaction.

For the case where part of the system interacts with another part (for example, considering the earth-moon system as a single object), then it makes sense to talk of equilibrium, and forces summing to zero, etc. etc just like conservation of energy. But in general, this cannot be done.

10. May 11, 2010

### Phrak

Andy, with your last paragraph, are you saying that force is not always conserved? I can't tell.

I sympathize with Russ. This is a very alien idea, and never talked aboutâ€”therefore suspect. The mathematics supports it, I'm sure, but I'm still trying to think of ways to break it. Did you find a way?

11. May 11, 2010

### Andy Resnick

There are 'conservative' forces, but not all forces are conservative forces- friction, for example. Conservative forces are derivable from a potential.

Newton's third law F12 = -F21 should be considered a definition of *contact* forces, not as a general property of 'force'- confusing the two is what leads to the example of pulling a cart and finding that no matter how hard you pull, the cart can't move.

12. May 11, 2010

### Studiot

I don't think the concept of conservation is applicable to forces.

Conservation of a property implies that there is a flow of that property, perhaps a fixed fund that can be 'used up' or 'transferred' to another body or location.
Forces are not susceptible to such limits.

I can connect a load to a battery and draw a current only a limited number of times.
If I bring a test mass near another body it experiences a force of attraction; I can remove the test mas and reintroduce it as often as I want, there will always be the force.

13. May 11, 2010

### D H

Staff Emeritus
Correct. This has nothing to do with Newton's third law, however.

The Earth and Moon are not in contact with one another, yet the gravitational force exerted by the Earth on the Moon is equal but opposite to the gravitational force exerted by the Moon on the Earth.

Using Newton's third law to draw the erroneous conclusion that no matter how hard you pull the cart, the cart can't move, is a common misinterpretation of Newton's third law. Newton's third law applies to two different bodies. The force exerted by object a on object b is equal but opposite to the force exerted by object b on object a. The forces exerted by object b on other objects has nothing to do with the motion of object b.

The weak form of Newton's third law follows from conservation of linear momentum. Add in conservation of angular momentum and you get the strong form of Newton's third law. The derivations implicitly assume that no momentum is stored in the fields that generate these forces. This is not always true; Newton's third law does not hold in such domains.

Ignoring this, if you want to look at Newton's third law as meaning force is conserved, fine. That is a rather non-standard view of the meaning of the word 'conserved'. I'm going to side with Russ, "The word 'conserved', though, doesn't really apply."

For a deeper explanation of why conservation of force doesn't really apply here, one must look to Noether's theorems. (Wikipedia reference because I don't have time to find a better one.)

14. May 11, 2010

### physicsworks

Actually Newton's third law is not always true, so the "force conservation" due to the third law is something odd to me, the same goes for the concept "force conservation" itself.

UPDATE
Sorry, I've missed this point.

Last edited: May 11, 2010
15. May 11, 2010

### Andy Resnick

I was unaware that there is a debate, especially since everyone is in agreement...?

16. May 12, 2010

### Phrak

Well, I'm not in agreement with some posters. Force really is conserved. It's just not very interesting to say that in homogeneous space

$$\frac{d}{dt}\sum 0 = 0 \ .$$

This doesn't make it false or an unreal symmetry. In fact, it should be a more general statement where translational symmetry is not an rule and momentum is not conserved within an interactive system.

Last edited: May 12, 2010
17. May 12, 2010

### Andy Resnick

My mistake.

Force is not a conserved quantity, for the reasons enumerated above.

18. May 12, 2010

### Hologram0110

Force is really more of an instantaneous thing. At any given moment if there is always a force in equal and opposite directions but the size force can change.

To me, conserve implies is constant over time, but as people have said, forces can and do change with time so this is not really the case.

19. May 12, 2010

### Phrak

How do you justify this wand waving without showing that the mathematical proof I presented above to be false or not appicable?

20. May 12, 2010

### polaris12

I just wanna mention I lost track of what you guys were talking about a while ago. Yea, I don't get what you guys are debating anymore.

21. May 12, 2010

### D H

Staff Emeritus
The only hand waving was in that post. You assumed Newton's third law to prove Newton's third law. Newton's third law does not always hold. The Biot-Savart law, for example, violates Newton's third law. That shouldn't be all that surprising; Maxwell's equations are inconsistent with Newtonian mechanics. Aside: The original intent of the Michelson Morley experiment was to disprove the claims of those upstart E&M physicists. As we all know, that was not the outcome of that experiment.

22. May 12, 2010

### gabbagabbahey

Phrak, what if the system contains momentum-carrying fields? What happens to $\sum_{i} \textbf{p}_i$ when momentum is transferred from a field to a particle?

23. May 12, 2010

### Andy Resnick

Work it out yourself- find a function f = dp/dt that satisfies both conservation of momentum

$$\frac{Dp}{Dt} = \nabla \cdot T + F_{ext}$$

and conservation of force

$$\frac {Df}{Dt} = \nabla \cdot Q + G_{ext}$$

Where T is the stress tensor, F the external force, Q and G the analogs to T and F.

I think you will find you cannot, unless f = 0.

24. May 12, 2010

### Staff: Mentor

The sum of the force vectors is equal and opposte, but not conserved. Equal and opposite doesn't include the concept of invariant over time - as I pointed out, the forces do vary over time.

Heck, the equations you are using to argue it don't include a time range! Where is time in this equation?:

$$\sum \textbf{F} = 0$$
Again, the concept of "over time" doesn't appear in that equation and without it, the word "conserved" doesn't apply. What you are really saying is just that that equation is always true over a period of time - as opposed to 'this time-dependent quantity is always constant'.

The way you are trying to use the word, every algebraic equation could have the word applied to it. Does 1+1=2 mean 2 is conserved? Does a=f/m mean acceleration is conserved? Similarly, does E=.5mv^2 mean kinetic energy is conserved?

Last edited: May 12, 2010
25. May 13, 2010

### mikeph

The forces vary but their sum does not.

Applying Newton's 3rd law at every instant should give sum(Force vectors) = 0 at every time. Isn't this a stronger statement than normal conservation?