Difference between revisions of "Second variation"
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| − | + | A special case of the $n$-th [[Variation of a functional|variation of a functional]] (see also [[Gâteaux variation|Gâteaux variation]]), generalizing the concept of the second derivative of a function of several variables. It is used in the calculus of variations. According to the general definition, the second variation at a point $x_0$ of a functional $f(x)$, defined on a normed space $X$, is | |
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| + | $$ \delta^2 f (x_0, h) = \frac{d^2}{d t^2} f (x_0 + th) |_{t = 0} $$ | ||
If the first variation is zero, the non-negativity of the second variation is a necessary, and the strict positivity | If the first variation is zero, the non-negativity of the second variation is a necessary, and the strict positivity | ||
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| − | a sufficient, condition (under certain assumptions) for a local minimum of | + | $$ \delta^2 f (x_0, h) \geqslant \alpha \| h \|^2, \hspace{1em} \alpha > 0 $$ |
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| + | a sufficient, condition (under certain assumptions) for a local minimum of $f(x)$ at the point $x_0$. | ||
In the simplest (vector) problem of the classical calculus of variations, the second variation of the functional | In the simplest (vector) problem of the classical calculus of variations, the second variation of the functional | ||
Revision as of 09:16, 8 January 2014
A special case of the $n$-th variation of a functional (see also Gâteaux variation), generalizing the concept of the second derivative of a function of several variables. It is used in the calculus of variations. According to the general definition, the second variation at a point $x_0$ of a functional $f(x)$, defined on a normed space $X$, is
$$ \delta^2 f (x_0, h) = \frac{d^2}{d t^2} f (x_0 + th) |_{t = 0} $$
If the first variation is zero, the non-negativity of the second variation is a necessary, and the strict positivity
$$ \delta^2 f (x_0, h) \geqslant \alpha \| h \|^2, \hspace{1em} \alpha > 0 $$
a sufficient, condition (under certain assumptions) for a local minimum of $f(x)$ at the point $x_0$.
In the simplest (vector) problem of the classical calculus of variations, the second variation of the functional
 |
considered on the vector functions of class 
with fixed boundary values 
, 
, has the form
 | (*) |
 |
where 
denotes the standard inner product in 
, while 
, 
, 
are matrices with respective coefficients
 |
(the derivatives are evaluated at the points of the curve 
). It is expedient to consider the functional of 
defined by (*) not only on the space 
, but also on the wider space 
of absolutely-continuous vector functions with a square-integrable modulus of the derivative. In this case the non-negativity and strict positivity of the second variation are formulated in terms of the non-negativity and strict positivity of the matrix 
(Legendre condition) and the absence of conjugate points (Jacobi condition), which are necessary conditions for a weak minimum in the calculus of variations.
A study of the second variation for extremals which may or may not supply a minimum (but, as before, satisfy the Legendre condition) has been carried out in variational calculus in the large [1]. The most important result was the coincidence of the Morse index of the second variation with the number of points conjugate to 
on the interval 
[2].
References
| [1] | M. Morse, "The calculus of variations in the large" , Amer. Math. Soc. (1934) |
| [2] | J.W. Milnor, "Morse theory" , Princeton Univ. Press (1963) |
Second variation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Second_variation&oldid=18617