View source for Second variation

A special case of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083720/s0837201.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083720/s0837202.png" /> of a functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083720/s0837203.png" />, defined on a normed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083720/s0837204.png" />, is

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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083720/s0837207.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083720/s0837208.png" />.

In the simplest (vector) problem of the classical calculus of variations, the second variation of the functional

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considered on the vector functions of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083720/s08372010.png" /> with fixed boundary values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083720/s08372011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083720/s08372012.png" />, has the form

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083720/s08372015.png" /> denotes the standard inner product in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083720/s08372016.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083720/s08372017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083720/s08372018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083720/s08372019.png" /> are matrices with respective coefficients

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(the derivatives are evaluated at the points of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083720/s08372021.png" />). It is expedient to consider the functional of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083720/s08372022.png" /> defined by (*) not only on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083720/s08372023.png" />, but also on the wider space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083720/s08372024.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083720/s08372025.png" /> ([[Legendre condition|Legendre condition]]) and the absence of conjugate points ([[Jacobi condition|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|variational calculus in the large]] [[#References|[1]]]. The most important result was the coincidence of the [[Morse index|Morse index]] of the second variation with the number of points conjugate to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083720/s08372026.png" /> on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083720/s08372027.png" /> [[#References|[2]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Morse,   "The calculus of variations in the large" , Amer. Math. Soc.  (1934)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.W. Milnor,   "Morse theory" , Princeton Univ. Press  (1963)</TD></TR></table>