How a Lagrangian can represent a theory?

In summary, the Lagrangian is used to calculate the particle behavior in quantum field theory. It is determined by the symmetries and order of equations of motion of the theory.
  • #1
hugo_faurand
62
10
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
 
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  • #2
The lagrangian is determined by the symmetries and the order of equations of motion you want. You could call this "representing".
 
  • #3
haushofer said:
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 ?
 
  • #5
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.
 

Related to How a Lagrangian can represent a theory?

1. What is a Lagrangian?

A Lagrangian is a mathematical function that represents the dynamics of a physical system. It is used in classical mechanics and quantum field theory to describe the motion and interactions of particles.

2. How does a Lagrangian represent a theory?

A Lagrangian represents a theory by providing a set of equations that describe the behavior of a physical system. These equations can be solved to predict the future behavior of the system, making it a powerful tool for understanding and predicting the behavior of complex systems.

3. What is the role of a Lagrangian in classical mechanics?

In classical mechanics, a Lagrangian is used to describe the motion of particles and systems in terms of their positions, velocities, and accelerations. It takes into account the forces acting on the system and allows for the calculation of the system's motion over time.

4. How does a Lagrangian differ from a Hamiltonian?

A Hamiltonian is another mathematical function used in classical mechanics and quantum mechanics to describe the dynamics of a physical system. While a Lagrangian is expressed in terms of position and velocity, a Hamiltonian is expressed in terms of position and momentum. Additionally, a Hamiltonian can also include potential and kinetic energy terms, while a Lagrangian only includes kinetic energy terms.

5. Can a Lagrangian represent any physical system?

Yes, a Lagrangian can represent any physical system as long as it is a well-defined system with a finite number of degrees of freedom. This includes systems in classical mechanics, quantum mechanics, and even relativistic systems. However, constructing a Lagrangian for some systems may be more complex than others.

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