Second variation

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

== Motivation == Much of the calculus of variations relies on the first variation, which is a generalization of the first derivative to a functional.<ref name=brechtkenmanderscheid1991introduction>{{cite book |last=Brechtken-Manderscheid |first=Ursula |year=1991 |title=Introduction to the Calculus of Variations |chapter=5: The necessary condition of Jacobi}}</ref> An example of a class of variational problems is to find the function <math>y</math> which minimizes the integral

on the interval <math>[a, b]</math>; <math>J</math> here is a functional (a mapping which takes a function and returns a scalar). It is known that any smooth function <math>y</math> 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 (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 <math>y</math>. The Taylor series of the integrand of our variational functional about a nearby point <math>y + \varepsilon h</math> where <math>\varepsilon</math> is small and <math>h</math> is a smooth function which is zero at <math>a</math> and <math>b</math> is

<math display="block"> 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).</math>

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

<math display="block">\delta^2J(h, y) := \int_a^b h^2 f_{yy} + 2hh' f_{yy'} + f_{y'y'} h'^2.</math>

It can then be shown that <math>J</math> has a local minimum at <math>y_0</math> if it is stationary (i.e. the first variation is zero) and <math>\delta^2J(h, y_0) \geq 0</math> for all <math>h</math>.<ref name=vanbrunt2003calculus>{{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 }}</ref>

== The Jacobi necessary condition == === The accessory problem and Jacobi differential equation === {{distinguish|Hamilton–Jacobi equation}}

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

<math display=block>\Omega_h - \frac{d}{dx} \Omega_{h'} = 0.</math>

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

The '''Jacobi necessary condition''' then follows:

{{Blockquote| text=Let <math>y</math> be an extremal for a variational integral on <math>[a,b]</math>. Then a point <math>c \in (a, b)</math> is a conjugate point of <math>a</math> only if <math>f_{y'y'}(c, y, y') = 0</math>.<ref name=brechtkenmanderscheid1991introduction></ref>}}

In particular, if <math>f</math> satisfies the '''strengthened Legendre condition''' <math>f_{y'y'} > 0</math>, then <math>y</math> is only an extremal if it has no conjugate points.<ref name=vanbrunt2003calculus></ref>

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.<ref>{{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}}</ref>

== An example: shortest path on a sphere == {{See also | Great circle}} 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<ref name=brechtkenmanderscheid1991introduction></ref>

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

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

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 <math>b > \pi</math>, 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.

== References == {{reflist}}

== Further reading == {{ref begin}} *M. Morse, "The calculus of variations in the large" , Amer. Math. Soc. (1934) *J.W. Milnor, "Morse theory" , Princeton Univ. Press (1963) *Weishi Liu, Chapter 10. The Second Variation, University of Kansas [https://liu.ku.edu/Lectures%20Ch10.pdf] *Lecture 12: variations and Jacobi fields [http://staff.ustc.edu.cn/~wangzuoq/Courses/16S-RiemGeom/Notes/Lec12.pdf] {{ref end}}

Category:Calculus of variations