Chapter 5
Symmetries and Conservation Laws
At a very fundamental level, physics is about identifying patterns of order in Na- ture. The inception of the field starts arguably with Tycho Brahe (1546-1601) — the first modern experimental physicist — and Johannes Kepler (1571-1630) — the first modern theoretical physicist. In the 16th century, Brahe collected large amounts of astrophysical data about the location of planets and stars with groundbreaking accuracy — using an impressive telescope set up in a castle. Kepler pondered for years over Brahe’s long tables of numbers until he could finally iden- tify a pattern underlying planetary dynamics: this was summarized in the three laws of Kepler. Later on, Isaac Newton (1643-1727) referred to these achieve- ments through his famous quote: “If I have seen further it is only by standing on the shoulders of giants”. Since then, physics has always been about identifying patterns in numbers, in measurements. And a pattern is simply an indication of a repeating rule, a constant attribute in complexity, an underlying symmetry. In 1918, Emmy Noether (1882–1935) published a seminal work that clarified the deep relations between symmetries and conserved quantities in Nature. In a sense, this work organizes physics in a clear diagram and gives one a bird’s eye view of the myriad of branches of the field. Noether’s theorem, as it is called, can change the way one thinks about physics in general. It is profound yet simple. While one can study Noether’s theorem in the Newtonian formulation of me- chanics through cumbersome methods, the subject is an excellent demonstration of the power of the new formalism we developed, Lagrangian mechanics. In this sec- tion, we use the variational principle to develop a statement of Noether’s theorem. We then prove it and demonstrate its importance through examples.
189 190 CHAPTER 5. SYMMETRIES AND CONSERVATION LAWS
Figure 5.1: Two particles on a rail.
EXAMPLE 5-1: A simple example
Let us start with a simple mechanics problem from an example we saw in Section 4.1. We have two particles, with masses m1 and m2, confined to moving along a horizontal frictionless rail as depicted in Figure 5. The action for the system is
1 1 S = dt m q˙2 + m q˙2 V (q q ) (5.1) 2 1 1 2 2 2 1 2 Z ✓ ◆ where we are considering some interaction between the particles described by a potential V (q q ) that depends only on the distance between the particles. 1 2 Let us consider a simple transformation of the coordinates given by
q10 = q1 +C ,q20 = q2 + C (5.2) where C is some arbitrary constant. We then have
q˙10 =˙q1 , q˙20 =˙q2 . (5.3)
Hence, the kinetic terms in the action are unchanged under this transformation. Furthermore, we also have
q0 q0 = q q (5.4) 1 2 1 2 5.1. INFINITESIMAL TRANSFORMATIONS 191 implying that the potential term is also unchanged. The action then preserves its overall structural form under the transformation
1 2 1 2 S dt m q˙ + m q˙ V (q0 q0 ) . (5.5) ! 2 1 01 2 2 02 1 2 Z ✓ ◆ This means that the equations of motion, written in the primed transformed co- ordinates, are identical to the ones written in the unprimed original coordinates. We then say that the transformation given by (5.2) is a symmetry of our system. Physically, we are simply saying that — since the interaction between the particles depends only on the distance between them — a constant shift of both coordinates leaves the dynamics una↵ected. It is also useful to consider an infinitesimal version of such a transformation. Let us assume that the constant C is small, C ✏; and we write !
q0 q q = ✏ (5.6) k k ⌘ k for k =1, 2. We then say that qk = ✏ is a symmetry of our system. To make these ideas more useful, we want to extend this example by considering a general class of interesting transformations that we may want to consider.
5.1 Infinitesimal transformations
There are two useful types of infinitesimal transformations: direct and indirect ones.
5.1.1 Direct transformations A direct transformation deforms the degrees of freedom of a setup directly: