Second variation

In the calculus of variations, the second variation extends the idea of the second derivative test to functionals. Much like for functions, at a stationary point where the first derivative is zero, the second derivative determines the nature of the stationary point; it may be negative, positive or zero.
Via the second functional, it is possible to derive powerful necessary conditions for solving variational problems, such as the Legendre–Clebsch condition and the Jacobi necessary condition detailed below.

Motivation

Much of the calculus of variations relies on the first variation, which is a generalization of the first derivative to a functional. An example of a class of variational problems is to find the function which minimizes the integral
on the interval ; here is a functional. It is known that any smooth function which minimizes this functional satisfies the Euler-Lagrange equation
These solutions are stationary, but there is no guarantee that they are the type of extremum desired. A test via the second variation would ensure that the solution is a minimum.

Derivation

Take an extremum. The Taylor series of the integrand of our variational functional about a nearby point where is small and is a smooth function which is zero at and is
The first term of the series is the first variation, and the second is defined to be the second variation:
It can then be shown that has a local minimum at if it is stationary and for all.

The Jacobi necessary condition

The accessory problem and Jacobi differential equation

As discussed above, a minimum of the problem requires that for all ; furthermore, the trivial solution gives. Thus consider can be considered as a variational problem in itself - this is called the accessory problem with integrand denoted. The Jacobi differential equation is then the Euler-Lagrange equation for this accessory problem:

Conjugate points and the Jacobi necessary condition

As well as being easier to construct than the original Euler-Lagrange equation the Jacobi equation also expresses the conjugate points of the original variational problem in its solutions. A point is conjugate to the lower boundary if there is a nontrivial solution to the Jacobi differential equation with.
The Jacobi necessary condition then follows:
In particular, if satisfies the strengthened Legendre condition, then is only an extremal if it has no conjugate points.
The Jacobi necessary condition is named after Carl Jacobi, who first utilized the solutions for the accessory problem in his article, and the term 'accessory problem' was introduced by von Escherich.

An example: shortest path on a sphere

As an example, the problem of finding a geodesic between two points on a sphere can be represented as the variational problem with functional
The equator of the sphere, minimizes this functional with ; for this problem the Jacobi differential equation is
which has solutions. If a solution satisfies, then it must have the form. These functions have zeroes at, and so the equator is only a solution if.
This makes intuitive sense; if one draws a great circle through two points on the sphere, there are two paths between them, one longer than the other. If, then we are going over halfway around the circle to get to the other point, and it would be quicker to get there in the other direction.