# What is actually a conservation law?

Though in his lectures, Feynman didn't define conservation law, he did use it while explaining divergence theorem:

[...] heat is conserved. That is, no heat is generated inside the material and the amount that decreases inside equals the amount that goes outside [...] So, in circumstances in which heat would be conserved , we say that $$\int_s \mathbf{h} \cdot \mathbf{n} da = -\dfrac{dQ}{dt}.$$

Now, what is new in it? If something goes outside, then it will decrease inside! I can understand the equation but unable to concieve what really is "conservation-law".

While reading the wikipedia, other sites, some answers to my question here, I got to know that conservation deals with those stuff that doesn't appear to any other place, doesn't get destroyed, etc, etc. . .

What is the meaning of "conservation" actually ? And also, why is the equation termed as continuity equation ? What is meanty by "continuity"?

• Are you asking what it means to say that something is conserved? Or are you asking where conservation laws come from? I don't fully understand your question – Jim Mar 30 '15 at 15:25
• A continuity equation describes how a quantity stays constant when it is conserved. To move from one place to another, there must be continuous flow of the quantity. It's called a continuity equation because it shows the continuous flow; the continuity of the conserved quantity – Jim Mar 30 '15 at 15:50
• @Jimnosperm: Sir, I want to know what it means when a thing is conserved. – user36790 Mar 30 '15 at 16:05

When we say something is conserved or that there is a conservation law for a given thing, we mean that the quantity of it does not change. You neither lose nor gain any of that thing.

More specifically, conservation can come in two flavours.

Something can be globally conserved. This means that the total amount of that something in the universe does not change over time. You can measure the total amount today and every day after that, you can be sure the amount hasn't changed. It may seem silly and you may ask "Well what isn't conserved then?" but most things aren't conserved. The number of humans is not a globally conserved quantity. There's more of us now than ever before. But this is a rather weak type of conservation law. Something that is globally conserved only has to maintain the same amount across the whole universe. A global conservation does not inhibit something from being destroyed from one location and simultaneously created in another.

A stronger statement is that something is locally conserved. This means that the amount of it in any localized area is constant. The only way for the amount of that something to change in any given small volume is if some of it physically moves out of the region. You can't have any popping in and out of existence, just moving from place to place.

I should mention that a global conservation should always have a corresponding local conservation. Thanks to relativity, the only way to maintain global conservation is to maintain local conservation. However, a locally conserved value does not need to be globally conserved. It's possible to have the case where something is conserved at all small scales, but on scales large enough such that they can't be considered local, they may cease to be conserved. There are a number of ways this could happen, but the first that comes to mind is if the change in quantity of something is below the limit of the Heisenberg uncertainty on all local scales but above that limit on larger, global scales. But no matter the mechanism, the point is that global implies local, but local does not imply global. A good example is energy-momentum conservation, which is conserved locally, but not globally.

A continuity equation is how we describe movement under a local conservation law. It defines how something exists at every point as it moves along and, thus, how the local conservation law is upheld. It's called "continuity equation" because it's an equation that describes the continuous existence of the quantity as it flows from one place to another.

So to sum up, saying something is conserved means the amount in a given region does not change. Saying it is locally conserved means the amount in a small region stays constant and the only way to change it is to physically move the quantity to another place. Saying it is globally conserved means the total amount of something in the universe never changes.

• +1. Let me sum up: A locally conserved quantity can't be created or destroyed; either we have to put in(out) to increase(decrease) its quantity. Globally conserved quantity is also locally conserved like energy which means we can't teleport energy from earth to some other planet as it would hamper the continuity-flow. Continuity equation measures the continuous flow as that quantity can't disappear amidst the flow. Right? – user36790 Mar 31 '15 at 4:40
• @user36790 That's pretty much it, yes. Except that local conservation doesn't require that the total amount in the universe is conserved, just the total amount in a small area. Global conservation is what requires the total amount in the universe is conserved. But you definitely seem to have a good grasp on it – Jim Mar 31 '15 at 13:20

When you say

If something goes outside, then it will decrease inside!

what you assume is exactly a conservation law. It may seem trivial, but it is not necessarily.

Consider the population of a city, for example. At one point in time, you measure how many people are within the city borders; let's call this number $N_0$. Then, you observe all city boundaries and count how many people leave the city during your observation time, that number being called $N_{\textrm{left}}$, and how many enter the city during the same observation time, $N_{\textrm{entered}}$. After your observation time, you again count the number of people in the city, obtaining $N_1$.

Now, the equation $$N_1 = N_0 - N_{\textrm{left}} + N_{\textrm{entered}}$$ is exactly what one may call conservation of people. However, such a conservation law may not exist, depending on the model you are trying to describe: If you allow for people to be born or people to die, the equation above does not have to hold, and there would be no conservation law for people: People could leave the city, and still the number of people in the city could be higher after your observation interval than before.

• A biologist, a physicist and a mathematician observe two people enter an empty house, and three people leave. The biologist concludes that they reproduced. The physicist concludes that a conservation law has been violated. The mathematician concludes that when one person enters the house it will be empty again. – Eric Lippert Mar 30 '15 at 21:00

It has a very simple yet important meaning.It simply means that the quantity that you are observing will always stay the same,even if that means that it gets transferred to another form or convert to another medium.You can not simply create more "stuff" of that quantity and you can not destroy it.It can not be created from nothing and it can not just be eliminated from the whole universe.

• +1: . . .created from nothing is the phrase that greatly reflects the intuition of conservation-law. – user36790 Mar 31 '15 at 14:21