How a Lagrangian can represent a theory?

[hugo_faurand]

Hello everyone !

I recently read an article about Standard Model's Lagrangian. And it made me remember another article (that I read a long time ago) which said that a theory's Lagrangian "represent" the theory.

But How ?

Maybe I didn't get the sense of "represent".
Also I know that there is something called the Lagrangian density so maybe it refers to this.


Thanks in advance.

Regards

 

[haushofer]

The lagrangian is determined by the symmetries and the order of equations of motion you want. You could call this "representing".

 

[hugo_faurand]

The lagrangian is determined by the symmetries and the order of equations of motion you want. You could call this "representing".

Could you develop what is equatuond of motion's symmetries ?

 

[fresh_42]

Maybe an example can help:
https://www.physicsforums.com/threads/math-challenge-november-2018.960003/#post-6087946 (#23)
https://www.physicsforums.com/threads/math-challenge-november-2018.960003/page-3#post-6090980

 

[DEvens]

The Lagrangian is used in the path integral formulation of quantum field theory. Basically, there is a direct relation between terms in the Lagrangian and particle behavior. Google up Feynman diagrams.

https://en.wikipedia.org/wiki/Feynman_diagramhttps://simple.wikipedia.org/wiki/Feynman_diagram
Each term in the Lagrangian corresponds to something represented by a Feynman diagram, which represents something happening in the particle theory. Some of the symmetries are represented explicitly. Relativistic in-variance is built in by using the mu-nu 4-derivatives, and building things up out of fields that are relativistic vectors, tensors, etc. Gauge symmetry is built up from the symmetries of the gauge field, usually called A-sub-mu or some such.

To get into this more you will need a lot of math. You could start with learning the Hamiltonian-Lagrange formulation of classical mechanics. The intro to that was a 4 month course for me. After that you could step over to the Hamiltonian-Lagrange formulation of quantum mechanics, then of quantum field theory. That was another two 4-month classes for me.