Second variation

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In the calculus of variations, the second variation extends the idea of the second derivative test to functionals.^[1] 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 (if the point is a maximum point), positive (if a minimum) or zero (if a saddle point).

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.^[2]

Much of the calculus of variations relies on the first variation, which is a generalization of the first derivative to a functional.^[3] An example of a class of variational problems is to find the function ${\displaystyle y}$ which minimizes the integral

$${\displaystyle J[y]=\int _{a}^{b}f(x,y,y')dx}$$

on the interval ${\displaystyle [a,b]}$; ${\displaystyle J}$ here is a functional (a mapping which takes a function and returns a scalar). It is known that any smooth function ${\displaystyle y}$ which minimizes this functional satisfies the Euler-Lagrange equation

$${\displaystyle f_{y}-{\frac {d}{dx}}f_{y'}=0.}$$

These solutions are stationary, but there is no guarantee that they are the type of extremum desired (completely analogously to the first derivative, they may be a minimum, maximum or saddle point). A test via the second variation would ensure that the solution is a minimum.

Take an extremum ${\displaystyle y}$. The Taylor series of the integrand of our variational functional about a nearby point ${\displaystyle y+\varepsilon h}$ where ${\displaystyle \varepsilon }$ is small and ${\displaystyle h}$ is a smooth function which is zero at ${\displaystyle a}$ and ${\displaystyle b}$ is

$${\displaystyle f(x,y,y')=f(x,y,y')\varepsilon (hf_{y}+h'f_{y'})+{\frac {\varepsilon ^{2}}{2}}(h^{2}f_{yy}+2hh'f_{yy'}+h'^{2}f_{y'y'})+O(\varepsilon ^{3}).}$$

The first term of the series is the first variation, and the second is defined to be the second variation:

$${\displaystyle \delta ^{2}J(h,y):=\int _{a}^{b}h^{2}f_{yy}+2hh'f_{yy'}+f_{y'y'}h'^{2}.}$$

It can then be shown that ${\displaystyle J}$ has a local minimum at ${\displaystyle y_{0}}$ if it is stationary (i.e. the first variation is zero) and ${\displaystyle \delta ^{2}J(h,y_{0})\geq 0}$ for all ${\displaystyle h}$.^[4]

As discussed above, a minimum of the problem requires that ${\displaystyle \delta ^{2}J(h,y_{0})\geq 0}$ for all ${\displaystyle h}$; furthermore, the trivial solution ${\displaystyle h=0}$ gives ${\displaystyle \delta ^{2}J(h,y_{0})=0}$. Thus consider ${\displaystyle \delta ^{2}J(h,y_{0})}$ can be considered as a variational problem in itself - this is called the accessory problem with integrand denoted ${\displaystyle \Omega }$. The Jacobi differential equation is then the Euler-Lagrange equation for this accessory problem:^[5]

$${\displaystyle \Omega _{h}-{\frac {d}{dx}}\Omega _{h'}=0.}$$

As well as being easier to construct than the original Euler-Lagrange equation (due ${\displaystyle h}$ and ${\displaystyle h'}$ being at most quadratic) the Jacobi equation also expresses the conjugate points of the original variational problem in its solutions. A point ${\displaystyle c}$ is conjugate to the lower boundary ${\displaystyle a}$ if there is a nontrivial solution ${\displaystyle h}$ to the Jacobi differential equation with ${\displaystyle h(a)=h(c)=0}$.

The Jacobi necessary condition then follows:

Let ${\displaystyle y}$ be an extremal for a variational integral on ${\displaystyle [a,b]}$. Then a point ${\displaystyle c\in (a,b)}$ is a conjugate point of ${\displaystyle a}$ only if ${\displaystyle f_{y'y'}(c,y,y')=0}$.^[3]

In particular, if ${\displaystyle f}$ satisfies the strengthened Legendre condition ${\displaystyle f_{y'y'}>0}$, then ${\displaystyle y}$ is only an extremal if it has no conjugate points.^[4]

The Jacobi necessary condition is named after Carl Jacobi, who first utilized the solutions for the accessory problem in his article Zur Theorie der Variations-Rechnung und der Differential-Gleichungen, and the term 'accessory problem' was introduced by von Escherich.^[6]

As an example, the problem of finding a geodesic (shortest path) between two points on a sphere can be represented as the variational problem with functional^[3]

$${\displaystyle J[y]=\int _{0}^{b}{\sqrt {\cos ^{2}y+y'^{2}}}dx.}$$

The equator of the sphere, ${\displaystyle y=0}$ minimizes this functional with ${\displaystyle f_{y'y'}=1>0}$; for this problem the Jacobi differential equation is

$${\displaystyle h''+h=0}$$

which has solutions ${\displaystyle h=A\sin(x)+B\cos(x)}$. If a solution satisfies ${\displaystyle h(0)=0}$, then it must have the form ${\displaystyle h=A\sin(x)}$. These functions have zeroes at ${\displaystyle k\pi ,k\in \mathbb {Z} }$, and so the equator is only a solution if ${\displaystyle b<\pi }$.

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 ${\displaystyle b>\pi }$, 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.