lagrangian - An example from classical mechanics, Lagrangians and Lagrangian densities in field theory, Electromagnetic Lagrangian

The difference between kinetic energy K and potential energy V; symbol L, units J (joule); L = K ? V; after Joseph Lagrange. It is the fundamental expression of the properties of a mechanical system, from which equations of motion can be derived using Euler–Lagrange equations.

A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. The equations of motion are obtained by means of an action principle, written as

where the action is a functional

denoting the set of parameters of the system.

The Lagrange formulation of mechanics is important not just for its broad applications (see below) but also for its role in advancing deep understanding of physics. Although Lagrange sought to describe classical mechanics, the action principle that is used to derive the Lagrange equation is now recognized to be deeply tied to quantum mechanics: physical action and quantum-mechanical phase (waves) are related via Planck's constant, and the principle of stationary action can be understood in terms of constructive interference of wave functions.

An example from classical mechanics

The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics known as Lagrangian mechanics.

Suppose we have a three dimensional space and the Lagrangian

Then, the Euler-Lagrange equation is where the time derivative is written conventionally as a dot above the quantity being differentiated, and is the del operator.

Using this result we can easily show that the Lagrangian approach is equivalent to the Newtonian one.

Suppose we have a three-dimensional space in spherical coordinates, r, θ, φ with the Lagrangian

Then the Euler-Lagrange equations are:

Here the set of parameters is just the time , and the dynamical variables are the trajectories of the particle.

Lagrangians and Lagrangian densities in field theory

In field theory, occasionally a distinction is made between the Lagrangian L, of which the action is the time integral

and the Lagrangian density , which one integrates over all space-time to get the action:

The Lagrangian is then the spatial integral of the Lagrangian density.

Electromagnetic Lagrangian

Generally, in Lagrangian mechanics, the Lagrangian is equal to

L = T − V

where T is kinetic energy and V is potential energy. Given an electrically charged particle with mass m and charge q, with velocity v in an electromagnetic field with scalar potential φ and vector potential A, the particle's kinetic energy is

and the particle's potential energy is

where c is the speed of light. Then the electromagnetic Lagrangian is

Lagrangians in Quantum Field Theory

Note that in the following, ħ = c = 1.

Dirac Lagrangian

The Lagrangian density for a Dirac field is

where ψ is a spinor, is its Dirac adjoint, is the gauge covariant derivative, and is Feynman notation for .

Quantum Electrodynamic Lagrangian

The Lagrangian density for QED is

where Fμν is the electromagnetic tensor

Quantum Chromodynamic Lagrangian

The Lagrangian density for quantum chromodynamics is

where is the QCD gauge covariant derivative, and Fαμν is the gluon field strength tensor.

Before we go on, let's give some examples:

In classical mechanics, in the Hamiltonian formalism, M is the one dimensional manifold , representing time and the target space is the cotangent bundle of space of generalized positions. In other words,

Most of the time, we will also assume in addition that the Lagrangian depends on only the field value and its first derivative but not the higher derivatives;

Given boundary conditions, basically a specification of the value of φ at the boundary if M is compact or some limit on φ as x approaches (this will help in doing integration by parts), the subspace of consisting of functions, φ such that all functional derivatives of S at φ are zero and φ satisfies the given boundary conditions is the subspace of on shell solutions.

The solution is given by the Euler-Lagrange equations (thanks to the boundary conditions),

Incidentally, the left hand side is the functional derivative of the action with respect to φ.