Submission declined on 11 March 2024 by The Herald (talk).
Where to get help
How to improve a draft
You can also browse Wikipedia:Featured articles and Wikipedia:Good articles to find examples of Wikipedia's best writing on topics similar to your proposed article. Improving your odds of a speedy review To improve your odds of a faster review, tag your draft with relevant WikiProject tags using the button below. This will let reviewers know a new draft has been submitted in their area of interest. For instance, if you wrote about a female astronomer, you would want to add the Biography, Astronomy, and Women scientists tags. Editor resources
|
Submission declined on 13 January 2024 by Sungodtemple (talk). This submission is not adequately supported by reliable sources. Reliable sources are required so that information can be verified. If you need help with referencing, please see Referencing for beginners and Citing sources. Declined by Sungodtemple 5 months ago. |
- 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 (talk • contribs) 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
editMuch 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
editTake 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
editThe accessory problem and Jacobi differential equation
editAs 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
editAs 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
editAs 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
edit- ^ "Second variation". Encyclopedia of Mathematics. Springer. Retrieved January 14, 2024.
- ^ "Jacobi condition". Encyclopedia of Mathematics. Springer. Retrieved January 14, 2024.
- ^ a b c Brechtken-Manderscheid, Ursula (1991). "5: The necessary condition of Jacobi". Introduction to the Calculus of Variations.
- ^ 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.
- ^ "Jacobi Differential Equation". Wolfram MathWorld. Retrieved January 12, 2024.
- ^ Bliss, Gilbert Ames (1946). "I.11: A second proof of Jacobi's condition". Lectures on the Calculus of Variations.
- in-depth (not just passing mentions about the subject)
- reliable
- secondary
- independent of the subject
Make sure you add references that meet these criteria before resubmitting. Learn about mistakes to avoid when addressing this issue. If no additional references exist, the subject is not suitable for Wikipedia.