# What exactly makes a force conservative?

I get that forces can be classified as either conservative or non-conservative, depending on whether the work done in a round trip is zero or non-zero.

What property of the force makes it to be, conservative or non-conservative, so that the work done in a round trip is zero/non-zero?

Note: I'm not asking the conditions for a force to be conservative. I'm asking what exactly makes it conservative.

• What do you mean by "what makes it conservative" if not the definition of conservative? – Javier Sep 15 '17 at 13:36
• When only conservative forces are involved, mechanical energy of a system is conserved but when non-conservative forces are involved energy is lost as heat. In a conservative force field like gravity, kinetic energy and potential energy are interconvertible. – Mitchell Sep 15 '17 at 13:38
• Related: physics.stackexchange.com/q/32052/2451 , physics.stackexchange.com/q/31672/2451 and links therein. – Qmechanic Sep 15 '17 at 14:17
• No one seems to be bringing up symmetry and Noether's theorem – Hovercraft Full Of Eels Sep 15 '17 at 23:40
• @HovercraftFullOfEels So write an anwer explaining how it applies. – sammy gerbil Feb 2 '18 at 17:13

All fundamental forces are conservatives and I would say that this is a postulate. Fundamental physics is constructed in such way that there is a quantity called energy which can be assigned to every possible state. If any fundamental process seems to violate conservation of energy we nowadays believe that there are some states, processes or even interactions that we are missing to take into account. Once we are able to take into account every state and interaction, the system and its interactions are conservative.

On the other hand, at macroscopic level, most of times we are not able to describe the system in terms of fundamental forces. We need to replace the zillions of coupled equations describing the dynamics of the system by a single equation or force, which we shall call effective, and which can describe the macroscopic results we observe. However, in this process we may miss many of the states and processes occurring such that we are no longer able to keep track of the mechanical energy balance. Energy balance would fail unless we consider other forms of energy, such as heat, which is also an effective quantity. A classic example is friction. We are not able to describe two macroscopic surfaces interacting in terms of every microscopic particle participating in the process. So we forget about it and assume there is an effective force called friction. Mechanical energy balance fails and we need to assume that the effective missing energy is present in the form of heat. That is why friction is non conservative. Another example is that of an time varying potential. It is only non conservative because we are effectively replacing a large, with many particles and closed system by one small, with few particles under external interaction. There is something that we are not able to keep track whose effect is the same as of a time varying potential.

• A time changing magnetic field creates a non conservative electric field, which exerts a non conservative force on charged particle (picture a betatron). So how is this conservative? – lalala Sep 15 '17 at 14:11
• @lalala definition of fundamental forces hyperphysics.phy-astr.gsu.edu/hbase/Forces/funfor.html – anna v Sep 15 '17 at 15:41
• @lalala You need to include in the system the force that is changing the magnetic field. – Ian Sep 15 '17 at 16:45
• @Ian: That still doesn't make the electric force any more conservative though. – Mehrdad Sep 16 '17 at 12:05

As you know, energy is always conserved.

When we talk of a force being non-conservative, it means that the force is operating within a system from which energy is allowed to escape.

Perhaps the most common example of this is a system where work is being done in the presence of friction. We talk of work being useful or not and that defines a parameter of our system. Non-useful work, in the form of work that generates heat instead of mechanical action, falls outside of our system, so it is "lost". Thus, that force is non-conservative.

In short, non-conservative forces are an emergent property of systems that define a boundary for energy transfers under consideration. Conservative forces transfer energy within the system. Non-conservative forces transfer energy out of (or, indeed, into) the system.

This might seem like a restatement of the definition of conservative and non-conservative forces but I think it sheds some light on what exactly makes a force conservative or otherwise.

One first must realize that forces are mediated by fields (well, more precisely by the exchange of virtual particles, but let's keep it all classical for the sake of simplicity and the fact that I don't understand QFT ;)). Now, it might have been presented to you that fields are just mathematical fictions used to do calculations but they are just as real as anything else. They themselves contain momentum, energy, angular momentum, everything.

Now, the action of a force on a particle is really just the interaction of the particle with the field. It might happen that the nature of the interaction between the field and the particle is of such a nature that when the particle goes in a complete loop, the energy that the field has transferred to the particle is exactly zero. In such a case, we call the field to be conservative. If the particle comes back to the position in which it earlier was but during the process, the field has taken away some energy from it or has given some energy to it then we call that force-field to be non-conservative.

• -1 You are only restating what the OP said he already knows, in his first sentence. – sammy gerbil Feb 3 '18 at 20:53
• @sammygerbil With all due respect, I don't see a word about the interaction between the field and the particle in OP's first sentence. – Dvij Mankad Feb 4 '18 at 0:40
• I did not say there was. I said that your answer is merely restating the definition which the OP gave, that forces are conservative if work done during a round trip is zero, and non-conservative if work done during a round trip is non-zero. Saying that energy is exchanged with a 'field' does not tell the OP what property of the force (or field) explains why one force is conservative while another is non-conservative. – sammy gerbil Feb 4 '18 at 1:01
• @sammygerbil Yes, I apologize. I realize your point. Thank you for patiently explaining your point. I think you are correct that it doesn't really explain why it happens. It just states what happens with a bit more added detail that the lost energy goes into the field. – Dvij Mankad Feb 4 '18 at 1:30

Throw a ball up in the air. You have provided it with an initial velocity (how else will it go up) and therefore also imparted it Kinetic Energy (KE) right at the beginning. Observe the ball move up.

As the ball moves up, you see the velocity of the ball is reducing since the force of gravity is acting against it. In Physics, we say that this force of gravity is doing negative work on the ball.

The ball has now reached the top, its velocity is zero. Basically force of gravity has done enough negative work to reduce the velocity to zero and therefore its KE has also becomes zero. Let’s say this total work done by gravity in upward journey is W1 (it would be a negative sign e.g. -4J or -10J)

But what has happened to the initial KE. We have been told in smaller classes that energy can neither be created nor destroyed, it can only change form. This is true here as well. The KE of the ball keeps reducing as it moves up but another form of energy keeps increasing. This other form of energy is Potential Energy (PE). This energy is determined by the distance between the ball and the earth (often Physics books term it as earth - particle system configuration). Thus the PE at start is zero and keeps increasing till all KE has converted to PE by the time it reaches the top of the flight. The gravitational force that did negative work on the ball increased the PE of the ball.

Now imagine the downward path of the ball. It is ready to go back to earth because it has the PE stored. As it moves down, the force of gravity is helping it increase its velocity and therefore the KE. We say that the force of gravity is doing positive work on the ball (let us say it is W2 and it will be positive, like +4J or +10J etc.). The KE keeps increasing and the PE keeps reducing, till all PE is converted to KE when the ball hits the ground. You would have guessed that the velocity at which the ball hits the ground would be same as that at which it was projected.

Now in this entire journey of the ball, the force of gravity would be termed as a conservative force. But why?

A force would be termed a conservative force if it meets the following conditions-

1. The system consists of 2 or more objects. In this case it is ball and earth The force acts between these 2 objects
2. Force does work on the object when the configuration of the objects changes and transfers energy between KE and PE. It is W1 in this case and the change in configuration is nothing but the change in distance between the ball and earth.

When the configuration is reversed, the force reverse the energy transfer. In this case it is work W2 during the balls downward journey

When W1 + W2 = 0, in a closed loop path, we say that the force in play is a conservative force.

Thus is the force has done work of say -10 J in upward journey, it would have done a work of +10J in the downward journey

You may like to watch this video made by me to understand this concept deeper

Conservative Forces

• -1 This answer does not address the question : What property of the force makes it to be conservative or non-conservative? – sammy gerbil Feb 2 '18 at 17:18
• I have first tried to explain the concept. If you see the last few paragraphs, you'll find that they clearly spell out what properties make a force conservative or non conservative – Vish Feb 3 '18 at 18:18
• You have only explained what the OP already knows : I get that forces can be classified as either conservative or non-conservative, depending on whether the work done in a round trip is zero or non-zero. – sammy gerbil Feb 3 '18 at 18:30
• ok, sorry, I'll try better next time – Vish Feb 4 '18 at 13:53

## protected by Qmechanic♦Sep 15 '17 at 15:10

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