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.

The Langrangian is also very useful in that it greatly simplifies calculations of dynamical systems.

Table of contents

1 Examples from classical mechanics
2 Mathematical formalism
3 See also

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:

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.