What Are Examples of Lagrangians in Various Disciplines?

In summary: The Lagrangian density here is${\cal L} = \frac{1}{2}\partial_\mu\phi\partial^\mu\phi - \frac{1}{2}m^2 \phi^2$In summary, the Lagrangian is a function used in mechanics and other disciplines to describe the behavior of a system by taking into account its kinetic and potential energy. Examples of Lagrangians can be found in fields such as particle physics, general relativity, electromagnetism, economics, and quantum field theory. The Lagrangian is an important tool in understanding the dynamics of a system and can vary in complexity depending on the specific discipline it is applied to.
  • #1
JTC
100
6
Assuming generlized variables, q, we have a Lagrangian in mechanics as the kinetic energy, K, minus potential energy, U, with a dependency form such that

L(q,dq/dt) = K(q, dq/dt) - U(q)

Can someone provide examples of Lagrangians in other disciplines?
 
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  • #2
What do you mean by other disciplines?
 
  • #3
PeroK said:
What do you mean by other disciplines?

I VAGUELY recall a Lagrangian in particle physics that was about a page long.

Just other examples of Lagrangians...
 
  • #5
Last edited:
  • #6
JTC said:
I VAGUELY recall a Lagrangian in particle physics that was about a page long.
An example of a much simpler Lagrangian of that type, which is often taught as a stepping-stone to quantum field theory:

Classical Scalar Field in Four Dimensions

Here we use not the Lagrangian of an entire system, but rather the Lagrangian density at each spacetime point.
 

Related to What Are Examples of Lagrangians in Various Disciplines?

What is a Lagrangian?

A Lagrangian is a mathematical function used in classical mechanics that describes the dynamics of a system based on its position and velocity. It is named after the Italian mathematician and physicist Joseph-Louis Lagrange.

What is the purpose of a Lagrangian?

The purpose of a Lagrangian is to simplify the equations of motion for a system by using a single function instead of multiple equations. This allows for easier analysis and prediction of the system's behavior.

What are some examples of a Lagrangian?

Some common examples of a Lagrangian include the Lagrangian for a simple harmonic oscillator, the Lagrangian for a pendulum, and the Lagrangian for a rigid body rotating about a fixed axis.

How is a Lagrangian different from a Hamiltonian?

A Lagrangian and Hamiltonian are both mathematical functions used in classical mechanics, but they differ in the variables they use. A Lagrangian uses position and velocity, while a Hamiltonian uses position and momentum.

Are there any real-world applications of Lagrangian mechanics?

Yes, Lagrangian mechanics has many real-world applications, such as in engineering, physics, and astronomy. It is used to model and predict the behavior of systems ranging from simple pendulums to complex systems like the motion of planets in our solar system.

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