A special case of the -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 of a functional , defined on a normed space , is
If the first variation is zero, the non-negativity of the second variation is a necessary, and the strict positivity
a sufficient, condition (under certain assumptions) for a local minimum of at the point .
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 . 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 .
|||M. Morse, "The calculus of variations in the large" , Amer. Math. Soc. (1934)|
|||J.W. Milnor, "Morse theory" , Princeton Univ. Press (1963)|
Second variation. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Second_variation&oldid=18617