# Noether’s Theorem

The huge picture

This video was produced by The Science Asylum$$^{[1]}$$.

In this section we’ll discuss what the physicist Ransom Stephens described as the greatest contribution to humanity. Noether’s Theorem, in the simplest terms, can be explained as follows: For every symmetry, there is a corresponding conservation law. Given a symmetry, we can use Hamilton’s principle to pull the mathematical crank and get a law of nature. Feynman described the general character of the history of physics as discoveries of many phenomena, all of which, seemingly different, in fact merely being emergent phenomena and different aspects of only one underlying physical thing—perhaps the most famous example of this being the grand unification of electricity and magnetism, once thought of as separate, in fact being two different aspects of the same physical thing, namely electromagnetism. Stephen Weinberg’s insight amplifies the significance of this observation; he said that the underlying physical mechanisms which give rise to the myriad of phenomena are more simple and fundamental. Many modern theories in physics now treat space and time as an emergent phenomena of a simpler set of underlying physical mechanisms which are more fundamental. Indeed, much research—such as that done by Leonard Susskind, Sean Carrol, and others—are attempting to reconcile the laws of quantum mechanics and relativistic mechanics by treating space and time as emergent phenomena. I mention this to give a glimpse of how far this amalgamation process has progressed, but now we shall take a step back and ask the question: what gives rise to the laws of nature? Are the laws of nature themselves an emergent result of a deeper set of underlying physical mechanisms? They in fact are. As Dr. Ransom Stephens once explained and as we shall see in the mathematical demonstrations which follow, Noether’s theorem implies that the fabric and very nature of space and time shape the natural laws we observe in the universe$$^{1}$$. The mathematician David Hilbert was the first to notice these implications and lead him to independently formulate the general theory of relativity—the theory that gravity is due to the curvature of space and time. This was far more elegant and magnificent than even the great Einstein’s formulation of general relativity.

Spatial symmetry implies the conservation of momentum

Let's consider physical systems where the Lagrangian depends upon only the difference $$q_j-q_i$$ in their coordinate values. For example, if you generalized coordinates are chosen to represent the location $$r$$ of every particle, then the difference $$q_j-q_i$$ will represent the spatial separation distance between each pair of particles in the system. Isolated systems of massive particles held together by gravity or charged particles held together by electrical forces are examples of such systems since the potential energy function (for both such systems) depends upon only the separation distances between particles. We can represent this dependence by writing $$V(q_j-q_i)$$. The find the motion of such a system, we'll follow the same general procedure that we have been using for the problems in the previous sections: namely, evaluate the derivatives in the Euler-Lagrange equation which is given by

$$\frac{∂L}{∂q_j}=\frac{d}{dt}\frac{∂L}{∂q_j’}.\tag{1}$$

Let's take the sum on both sides of Equation (1) to get

$$\sum_j\frac{∂L}{∂q_j}=\sum_j\frac{d}{dt}\frac{∂L}{∂q_j’}.\tag{2}$$

Let's start out by evaluating the derivatives on the left-hand side to get

$$\sum_j\frac{∂L}{∂q_j}=\sum_j\frac{∂}{∂q_j}\biggl(\frac{mq_j'^2}{2}-V(q_j-q_i)\biggl).$$

All of the kinetic energy terms drop out and we'll be left with

$$\sum\frac{∂}{∂q_j}\biggl(-V(q_j-q_i)\biggl)=-V(q_j-q_i).\tag{3}$$

Now, let's evaluate the same partial derivatives above except with respect to $$q_i$$ to get

$$\sum\frac{∂}{∂q_j}\biggl(V(q_j-q_i)\biggl)=V(q_j-q_i).\tag{4}$$

If we add up Equations (3) and (4) we'll see that we'll get zero:

$$\sum\frac{∂}{∂q_j}\biggl(V(q_j-q_i)\biggl)+\sum\frac{∂}{∂q_j}\biggl(-V(q_j-q_i)\biggl)=0.$$

This means that if we take the derivative with respect to time of the above equation, we'll get zero. Let's plug that into Equation (2) to get:

$$\sum_j\frac{d}{dt}\frac{∂L}{∂q_j’}=0.\tag{5}$$

If we evaluate each of the partial derivatives $$\frac{∂L}{∂q_j’}$$ using Cartesian coordinates, Equation (5) will simplify to the law of conservation of momentum

$$\sum_j\frac{d}{dt}mq_j'=0.$$

The equation above can of course be rewritten in its conventional form as

$$\sum_j\frac{d}{dt}mq_j'=0.$$

2. It is easy to verify that the Lagrangian of an isolated system of $$n$$ particles is given by $$L=\sum_{j,i}\biggl(\frac{mq_j'^2}{2}-V(q_j-q_i)\biggl)$$. When we take the sum from $$j=1$$ to $$j=n$$ in Equation (1), if you wanted to you could "collect" and group together each $$\frac{mq_j'^2}{2}$$ term to get the total kinetic energy of the system. When we take the sum from $$i=1$$ and $$j=1$$ to $$i=n$$ and $$j=n$$ in Equation (1), we could collect every potential energy term and put it behind the minus sign to get the potential energy of the whole system (with the minus sign in front of it of course).