Lagrangian is basic formulation of Quantum Field Theory

Quantum field theory is a unification of special relativity and quantum mechanics. This theory formed the framework of the standard model in particle physics. Mathematical foundation in quantum field theory is the formulation of Lagrangian. We can observe a system by looking from its Lagrangian. Then by using Euler-Lagrange equation, the equation of motion of the system can be obtained. And many more works can be done from the Lagrangian.
Let us consider properties of the Lagrangian further. If the field has a kinetic energy T and the potential  V, then Lagrangian is

In the continuous case we actually work with the density of the Lagrangian

If we integrate the Lagrangian over time, we get a new quantity called the action S

Action is functional because the action always takes functions as arguments and produces a number. Particles always take the path with the smallest action. To find the path, then the variation of the action should be minimized. This is done by describing the action as the minimum mean value and a mean variation.

The action is minimum if satisfied

In quantum field theory, since that used is a lagrangian density, then the equation for the action will be

For the sake of abbreviation, the Lagrangian density is often called just Lagrangian.
This is an example of a Lagrangian for a free scalar particle

The first term is the kinetic energy (containing ) and the second term is the rate of mass (containing ).