Lagrangian

The Lagrangian density ( after the mathematician Joseph -Louis Lagrange ) plays a role in theoretical physics in the consideration of fields. Describes the density of the Lagrange function in a volume element. Therefore, the Lagrange function is defined as the integral of the Lagrangian with the observed volume:

With the observed field.

The real purpose of the Lagrangian density is the description of fields by equations of motion. So how do we obtain the Lagrange equations 2 species from the Hamilton 's principle, can the Lagrange equations for the fields from the Hamilton 's principle for fields obtained ( derivation ). According to equation of motion:

Example

A vibrating string in a dimension is obtained for the Lagrangian

In this example mean:

This Lagrangian density is

This results in the equation of motion of the vibrating string

Application in the theory of relativity

Common applications for the description of physical processes on the Lagrangian density instead of the Lagrangian function, especially in relativistic processes. Here is a covariant representation of the Lagrange function is desired, then the effect is

Defined. Thus the Lagrangian is a Lorentz scalar, invariant under Lorentz transformations:

• Field Theory
• Classical Mechanics