Draft:Second variation

  • Comment: No inline citations in #Motivation and #An example: shortest path on a sphere. Specific articles like this one should also be accessible to people who know little about the topic, needs more explanation and wikilinks.
    Notability remains to be seen; I have not looked at the books but the web source doesn't mention second variation. Sungodtemple (talkcontribs) 04:18, 13 January 2024 (UTC)

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]

Motivation

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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   which minimizes the integral

 

on the interval  ;   here is a functional (a mapping which takes a function and returns a scalar). 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 (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

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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 (i.e. the first variation is zero) and   for all  .[4]

The Jacobi necessary condition

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The accessory problem and Jacobi differential equation

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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[5]:

 

Conjugate points and the Jacobi necessary condition

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As well as being easier to construct than the original Euler-Lagrange equation (due   and   being at most quadratic) 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:

Let   be an extremal for a variational integral on  . Then a point   is a conjugate point of   only if  .[3]

In particular, if   satisfies the strengthened Legendre condition  , then   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]

An example: shortest path on a sphere

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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]

 

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.

References

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  1. ^ "Second variation". Encyclopedia of Mathematics. Springer. Retrieved January 14, 2024.
  2. ^ "Jacobi condition". Encyclopedia of Mathematics. Springer. Retrieved January 14, 2024.
  3. ^ a b c Brechtken-Manderscheid, Ursula (1991). "5: The necessary condition of Jacobi". Introduction to the Calculus of Variations.
  4. ^ a b van Brunt, Bruce (2003). "10: The second variation". The Calculus of Variations. Springer. doi:10.1007/b97436. ISBN 978-0-387-40247-5.
  5. ^ "Jacobi Differential Equation". Wolfram MathWorld. Retrieved January 12, 2024.
  6. ^ Bliss, Gilbert Ames (1946). "I.11: A second proof of Jacobi's condition". Lectures on the Calculus of Variations.