# “Find the Lagrangian of the theory”

I've heard a few of my professors throw around the term "finding the Lagrangian of a theory". What exactly is this referring to. From what I understand it seems that you determine invariances (symmetries) and they give you a hint for what your Lagrangian is. Furthermore there is more to the story because I know:

$L=T-U$ is only one of the forms the Lagrangian can take in classical mechanics. So far I only learned about the Lagrangian in classical mechanics and might be building up to a limited knowledge of Feynman's path integral in my QM course.

What other theories have Lagrangians and how you can tell?

Are all Lagrangians of a given theory equivalent?

• I suspect that, in this context, the phrase means find the free, self-interaction (if any), and interaction terms (if any) for the Lagrangian density of one or more quantum fields. The particular set of quantum fields and their interactions is the 'theory'. – Alfred Centauri Feb 7 '15 at 14:02

1. "Find the Lagrangian of the theory" typically means that you are given the (classical) equations of motion (EOMs) of some physical system, and are supposed to find the action functional $S$ so that the EOMs are (parts of) the Euler-Lagrange (EL) equations for $S$.

2. Note that an action principle/Lagrangian formulation does not always exists, cf. e.g. this Phys.SE post and links therein.

3. Two different actions that yield the same EL eqs. are called classically equivalent action formulations.

4. Adding total divergence terms in the action does not change the EL eqs., but are typically associated with other boundary conditions for the theory.

5. Quantum mechanically, in the the corresponding path integral formulation, two classically equivalent actions need not lead to equivalent quantum theories.

6. In fact, already the same classical action can give rise to non-equivalent quantum theories because of different quantization procedures, different operator ordering prescriptions, etc.

Usually the terms "Lagrangian" and "theory" can be considered the same. For a new theory, you have a new Lagrangian. For example, when we say "QED is different from QCD", we mean their Lagrangians are different. Each theory has its own Lagrangian.

Although, observable quantities (and especially the equation of motion) is more important than the Lagrangian. So, when they are invariant under Lagrangian changing, we say those transformations are the symmetries of the theory.

We have many theories in physics. For example Quantum electrodynamics (QED), Quantum chromodynamics (QCD), General relativity, New massive gravity (NMG), Topologically massive gravity (TMG), and so on! Finding the Lagrangian is not always simple and we have to consider many things. Symmetries and conservation laws are our hints to get the Lagrangian.