Lagrangian Noether's Theorem ❤️

In the previous sections, we have seen how the Lagrangian formalism can be used to describe the dynamics of a system. In this section, we will see how the Lagrangian can be used to derive conservation laws for a system. In particular, we will see how symmetries of the Lagrangian lead to conservation laws in the form of Noether's theorem.

There are a few reasons why this is important and interesting:

1. As Schneider puts it, it is "one of the most beautiful love stories in the universe". (Therefore, it also makes for a good pick-up line.)
2. It is also one example of a profound connection in physics that is just not apparent from Newton's laws.
3. This concept is also exceedingly important in quantum field theory, where symmetries play a central role in the formulation of the theory. The standard model is based around the symmetry group, and the conservation laws that arise from this symmetry are the conservation of electric charge, weak isospin, and color charge.

Table of Contents

• Table of Contents
• Symmetries and Conservation Laws
  • Free Particle
  • References

Symmetries and Conservation Laws

There is a deep connection between symmetries of a system and conservation laws. In particular, if a system has a symmetry, then there is a corresponding conservation law.

Here are a few examples of symmetries and their corresponding conservation laws:

Symmetry ❤️	❤️ Conservation Law
Time translation symmetry	Conservation of energy
Space translation symmetry	Conservation of momentum
Rotation symmetry	Conservation of angular momentum

Here are some more advanced examples:

Symmetry ❤️	❤️ Conservation Law
Lorentz symmetry	Conservation of four-momentum
Gauge symmetry	Conservation of electric charge
Scale symmetry	Conservation of dilatation current

In the context of the Lagrangian formalism, we can derive these conservation laws using Noether's theorem.

Formally, Noether's theorem mathematically states:

Noether's Theorem: If a system is defined by a Lagrangian , and under a continuous infinitesimal transformation , the Lagrangian changes as , then there exists a quantity :

such that .

We will start from the basics and derive this theorem from scratch.

Free Particle

Let's consider a particle of mass moving in a zero-force field, i.e., a free particle. The Lagrangian for this system is given by:

where is the potential energy of the particle. Notice that is a function of the position , and not the velocity . Since and , we have , which implies that is a constant. Therefore, we can write , where is a constant:

We shall take the system to be one-dimensional for simplicity. Recall from previous derivations that if the path of the particle is varied by a small amount , then the Lagrangian changes by:

Then, by the product rule, we saw that this gave:

The inner-left part of this equation, , equals , and is zero for the real path. We will call this part the equation of motion. We are now interested in the right part of the equation, . We shall now define a symmetry as:

Simple Symmetry: A symmetry is a transformation of the form , where is an arbitrary function of time, such that the Lagrangian is constant.

This is not the most general form of symmetry, but it is the simplest form that we will start with. Since the potential energy is constant, we have (as we derived earlier). Notice that this does not have an explicit dependence on (only ), and so that is one form of symmetry - spatial translation invariance. This occurs when , where is a constant that represents how much we are translating the system.

When a symmetry occurs, the Lagrangian is invariant under the transformation. This means that . For the real path, we already have the equation of motion as zero, so we are left with:

This means that the quantity is conserved. In other words, under spatial translation invariance, momentum is conserved. This is the first example of Noether's theorem in action.

References

• Schneider, Elliot (Physics with Elliot). "Symmetries & Conservation Laws: A (Physics) Love Story"