Lagrangian

In physics, a Lagrangian is a function designed to sum up a whole system; the appropriate domain of the Lagrangian is a phase space, and it should obey the so-called Euler-Lagrange equations. The concept was originally used in a reformulation of classical mechanics known as Lagrangian mechanics. In this context, the Lagrangian is commonly taken to be the kinetic energy of a mechanical system minus its potential energy. The concept has also proven useful as extended to quantum mechanics.

Table of contents

1 Mathematical formalism
2 Examples from classical mechanics
3 See also

Mathematical formalism

Suppose we have an n-dimensional manifold, M and a target manifold T. Let be the configuration space of smooth functions from M to T.

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

In classical mechanics, M is the one dimensional manifold , representing time and the target space is the tangent bundle of space of generalized positions.
In field theory, M is the spacetime manifold and the target space is the set of values the fields can take at any given point. For example, if there are m real-valued scalar fields, φ₁,...,φ_m, then the target manifold is . If the field is a real vector field, then the target manifold is isomorphic to . There's actually a much more elegant way using tangent bundles over M, but we will just stick to this version.

Now suppose there's a functional, , called the action. Note it's a mapping to , not . This has got to do with physical reasons.

In order for the action to be local, we need additional restrictions on the action. If , we assume S(φ) is the integral over M of a function of φ, its derivative and the position called the Lagrangian, . 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; this is only a matter of convenience, though, and is not true in general! We will make this assumption for the rest of this article.

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

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.

Examples from classical mechanics

Suppose we have a three dimensional space and the Lagrangian

Then, the Euler-Lagrange equation is where I have used the standard convention in classical mechanics of writing the time derivative as a dot above the thing being differentiated.

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

Then, the Euler-Lagrange equations are: