There is a deep relation between symmetries and conservation laws. By conservation we mean a quantity that stays constant and does not change over time. For example, if we have 10 indestructible balls that truly live up to that description, the number of balls will not change over time. It turns out that each symmetry in physics implies the existence of a quantity that is conserved in nature. That statement is a consequence of Noether’s Theorem , published in 1918, which holds that for every continuous symmetry there must exist a corresponding conservation law. Apart from being beautiful, symmetries, as discussed earlier in this chapter, play an indispensable role in physics.
For example, the translation symmetry in space involves the idea that, all other things being equal, experiments performed in different locations should lead to identical results. Yet that same symmetry has even broader consequences: it inexorably leads to the conservation of momentum (mass times velocity). That, in itself, is rather amazing given that the conservation of momentum is a much more complicated statement than the obvious fact that the outcome of physics experiments don’t depend on where we perform them.
These statements, in turn, can be used to reformulate Newtonian mechanics. Let us consider, for instance, the conservation of momentum for two particles, 1 and 2 \frac{d}{dt}(\vec{p_1}+\vec{p_2})=0
This differential equation shows that the change in momentum for the two particles is zero, meaning that total momentum is conserved–as it has to be, according to the laws of physics. We can then define the force on particle i to be F_i=\frac{d}{dt}\vec{p_i} In this way, we have recovered not only Newton’s Second Law, F=ma but his Third Law as well: From the above equations, we can see that the sum of the forces on particle 1 and particle 2 is zero, which is another way of saying that these two forces are equal and opposite.
Remember when we talked about how it is not always clear as to which laws in physics are fundamental? The modern view of physics holds that the conservation of momentum is more fundamental than Newton’s laws of motion because the former, though not the latter, is a direct consequence of a symmetry principle and that has a larger domain of applicability.
We are now starting to see the close connection between symmetry and invariance, or conservation. The three continuous symmetries we have discussed so far lead to the following conservations laws:
- Symmetry under space translation leads to the conservation of linear momentum.
- Symmetry under time translation leads to the conservation of energy.
- Symmetry under rotation leads to the conservation of angular momentum.
Puzzle
We have two containers, one holding green paint and the other holding white paint. Assume that the containers are the same size and contain exactly the same amount of paint. Suppose we scoop out a small quantity of green paint in one cup and put it in the white paint container. Then we scoop out the same quantity of paint from the mixed container and put it back in the green container (Fig. 12 ). What is higher? The concentration of green paint in the white paint container, or the concentration of white paint in the green paint container?
