second variation

In the calculus of variations, the second variation extends the idea of the second derivative test to functionals.{{cite web |title=Second variation |url=https://encyclopediaofmath.org/wiki/Second_variation |website=Encyclopedia of Mathematics |publisher=Springer |access-date=January 14, 2024}} 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.{{cite web |title=Jacobi condition |url=https://encyclopediaofmath.org/wiki/Jacobi_condition |website=Encyclopedia of Mathematics |publisher=Springer |access-date=January 14, 2024}}

Motivation

Much of the calculus of variations relies on the first variation, which is a generalization of the first derivative to a functional.{{cite book |last=Brechtken-Manderscheid |first=Ursula |year=1991 |title=Introduction to the Calculus of Variations |chapter=5: The necessary condition of Jacobi}} An example of a class of variational problems is to find the function $y$ which minimizes the integral

$J[y] = \int_a^b f(x, y, y')dx$

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

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

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.

Derivation

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

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

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

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

It can then be shown that $J$ has a local minimum at $y_0$ if it is stationary (i.e. the first variation is zero) and $\delta^2J(h, y_0) \geq 0$ for all $h$ .{{cite book |last=van Brunt |first=Bruce |date=2003 |title=The Calculus of Variations |url=https://link.springer.com/book/10.1007/b97436 |publisher=Springer |chapter=10: The second variation|doi=10.1007/b97436 |isbn=978-0-387-40247-5 }}

The Jacobi necessary condition

= The accessory problem and Jacobi differential equation =

As discussed above, a minimum of the problem requires that $\delta^2J(h, y_0) \geq 0$ for all $h$ ; furthermore, the trivial solution $h=0$ gives $\delta^2J(h, y_0) = 0$ . Thus consider $\delta^2J(h, y_0)$ can be considered as a variational problem in itself - this is called the accessory problem with integrand denoted $\Omega$ . The Jacobi differential equation is then the Euler-Lagrange equation for this accessory problem{{cite web |url=https://mathworld.wolfram.com/JacobiDifferentialEquation.html |title=Jacobi Differential Equation |website=Wolfram MathWorld |access-date=January 12, 2024}}:

$\Omega_h - \frac{d}{dx} \Omega_{h'} = 0.$

= Conjugate points and the Jacobi necessary condition =

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

The Jacobi necessary condition then follows:

{{Blockquote|

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

In particular, if $f$ satisfies the strengthened Legendre condition $f_{y'y'} > 0$ , then $y$ 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 [https://doi.org/10.1515/crll.1837.17.68 Zur Theorie der Variations-Rechnung und der Differential-Gleichungen], and the term 'accessory problem' was introduced by von Escherich.{{cite book |title=Lectures on the Calculus of Variations |first=Gilbert Ames |last= Bliss|year=1946|chapter=I.11: A second proof of Jacobi's condition}}

An example: shortest path on a sphere

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

$J[y] = \int_0^b \sqrt{\cos^2y + y'^2}dx.$

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

$h'' + h = 0$

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

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 $b > \pi$ , 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.