Noether’s Theorem

Emmy Noether was a woman who made significant contributions to the field of mathematics, most notably abstract algebra. She played a big role in defining the axioms of ring theory that we use today. Noether also worked on non-commutative algebras and linear transformations. Today, however, I will discuss a contribution she made in the field of physics, namely Noether’s  Theorem. Noether’s Theorem applies to Lagrangian mechanics and states that if the Lagrangian of a system is symmetric under the translation q→q*=qQ, and q’→q*’=q’Q’then the quantity (∂L/∂q’)Q’ is conserved. Let’s explore what this means.

A bit of background on Lagrangian mechanics is necessary to begin understanding Noether’s Thoerem. Usually when solving a physics problem, one would need to find all the forces on the object of concern. They then use Newton’s second law, ∑ F=ma, which allows them to find the second derivative of the object’s position, from which they could (hopefully)  find a function for the object’s position. This works well enough for simple physics problems such as a block sliding down a ramp without friction. However the problem increases in difficulty if we allow the ramp to slide on the surface it rests on, of course without friction. This is where Lagrangian mechanics comes in. The idea behind it is to minimize what is called the action, or Lagrangian. The Lagrangian of a system is simply equal to the kinetic energy of the system minus the potential energy of the system, usually written L=T-U, where T is kinetic energy, and U is potential energy.

In order to truly understand Lagrangian mechanics, one needs a rudimentary understanding of the Euler-Lagrange equations, which is derived using calculus of variations. Simply put, the Euler-Lagrange equations allow one to minimize what’s called a functional, or a function of functions which are usually expressed as an integral over a set of functions.  It can be shown that in maximizing the functional whose integrand is the Lagrangian of the system will give the equations of motion of a system. Therefore the Euler-Lagrange equations are applied to the Lagrangian of a system to give the system’s equation of motion. Though this method is more complicated, and almost not worth it for simple problems, it is extremely useful with more complicated systems. The Euler-Lagrange equations are as follows:

Euler-Lagrange

Where q is the set of coordinates which describe the system. I think we may finally be ready to look at what Noether’s Theorem says.

Noether’s Theorem states that if the Lagrangian of a system is symmetric about some translation q→q*=qQ, and q’→q*’=q’Q’, then a quantity is conserved, namely (∂L/∂q’)Q’, is conserved. Often times this tells us that the conserved quantity is the momentum of the system. We can see this because U is almost always (if not always) independent of q’ so the partial derivative of the Legrangian with respect to q’ is equal to the partial derivative of the kinetic energy with respect to q’.

The kinetic energy of an object is defined to be (1/2)m(q’)², thus the partial derivative with respect to q’ is  mq’, which is the definition of momentum. Let’s now go through a quick and simple example to help solidify our understanding.

Consider a block sliding down a ramp with no friction (the ramp is fixed in our case). Let us choose our coordinate system as follows

Ramp

where the arrows on the right hand side represent the forces, so we can ignore them, and the z-axis is pointing out of the page. Since motion will be restricted to the x and z directions (the block will not bounce or jump off the surface of the ramp), there will be no kinetic energy in the y direction. Thus our Lagrangian is

L(x,z,x’,z’)=(m/2)(x’²+z’²)-mg(xsinθ),

where the potential energy was found using the idea of virtual gravity, or the portion of gravity which points down the ramp. If we make the following transformations

x→x*=x+0

x’→x*’=x’

z→z*=z+C

z’→z*’=z’

where C is a constant, we can see that L(x,z,x’,z’)=L(x*,z*,x*’,z*’), so our Lagrangian is symmetric about the above transformation. Thus, Noether’s Theorem tells us that

(∂L/∂x’)(0)+(∂L/∂z’)C=constant,

since the first term is zero, this simplifies to

(∂L/∂z’)=A

Where A is the above constant divided by C. As mentioned above, (∂L/∂z’) is an expression for the object’s momentum in the z direction. Therefore momentum is conserved in the z direction.

This beautiful and very useful theorem is just one of the contributions Emmy Noether has made to the fields of math and physics. Its foolish that we follow tradition in believing that men are better at math than women. I mean, sure there may be more men who do math than women, especially in Noether’s day. But men not letting women do math, and telling women that they are bad at math from a young age, so they don’t see their own potential, is worlds away from women being inherently bad at math. Emmy Noether is one of the best counter examples to disprove the notion that women cannot do math as well as men.

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