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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
 
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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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083720/s0837209.png" /></td> </tr></table>
+
$$
 +
J (x) \  = \
 +
\int\limits _ {t _ 0} ^ {t _ 1}
 +
L (t,\  x,\  \dot{x} ) \  dt; \ \
 +
L: \  [t _{0} ,\  t _{1} ] \times
 +
\mathbf R ^{n} \times \mathbf R ^{n} \rightarrow \mathbf R ,
 +
$$
 +
 
 +
 
 +
considered on the vector functions of class  $  C ^{1} $
 +
with fixed boundary values  $  x( t _{0} ) = x _{0} $,
 +
$  x (t _{1} ) = x _{1} $,
 +
has the form
 +
 
 +
$$ \tag{*}
 +
\delta ^{2} J (x _{0} ,\  h) \  = \
 +
\int\limits _ {t _ 0} ^ {t _ 1}
 +
( \langle A (t) \dot{h} (t),\  \dot{h} (t) \rangle +
 +
$$
 +
 
 +
 
 +
$$
 +
+
 +
{} 2 \langle  B (t) \dot{h} (t),\  h (t)\rangle + \langle C (t) h (t) ,\  h (t)\rangle ) \  dt,
 +
$$
 +
 
  
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
+
where  $  \langle  \cdot ,\  \cdot \rangle $
 +
denotes the standard inner product in  $  \mathbf R ^{n} $,
 +
while  $  A(t) $,  
 +
$  B(t) $,  
 +
$  C(t) $
 +
are matrices with respective coefficients
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083720/s08372013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083720/s08372014.png" /></td> </tr></table>
+
\frac{\partial ^{2} L}{\partial \dot{x} \partial \dot{x}}
 +
,\ \ 
 +
\frac{\partial ^{2} L}{\partial x \partial \dot{x}}
 +
,\ \ 
 +
\frac{\partial ^{2} L}{\partial x \partial x}
  
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
+
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083720/s08372020.png" /></td> </tr></table>
 
  
(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.
+
(the derivatives are evaluated at the points of the curve $  x _{0} (t) $).  
 +
It is expedient to consider the functional of $  h $
 +
defined by (*) not only on the space $  C ^{1} $,  
 +
but also on the wider space $  W _{2} ^{1} $
 +
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 $  A(t) $([[
 +
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]]].
+
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 $  t _{0} $
 +
on the interval $  (t _{0} ,\  t _{1} ) $[[#References|[2]]].
  
 
====References====
 
====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>
 
<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>

Latest revision as of 20:03, 28 January 2020


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

$$ J (x) \ = \ \int\limits _ {t _ 0} ^ {t _ 1} L (t,\ x,\ \dot{x} ) \ dt; \ \ L: \ [t _{0} ,\ t _{1} ] \times \mathbf R ^{n} \times \mathbf R ^{n} \rightarrow \mathbf R , $$


considered on the vector functions of class $ C ^{1} $ with fixed boundary values $ x( t _{0} ) = x _{0} $, $ x (t _{1} ) = x _{1} $, has the form

$$ \tag{*} \delta ^{2} J (x _{0} ,\ h) \ = \ \int\limits _ {t _ 0} ^ {t _ 1} ( \langle A (t) \dot{h} (t),\ \dot{h} (t) \rangle + $$


$$ + {} 2 \langle B (t) \dot{h} (t),\ h (t)\rangle + \langle C (t) h (t) ,\ h (t)\rangle ) \ dt, $$


where $ \langle \cdot ,\ \cdot \rangle $ denotes the standard inner product in $ \mathbf R ^{n} $, while $ A(t) $, $ B(t) $, $ C(t) $ are matrices with respective coefficients

$$ \frac{\partial ^{2} L}{\partial \dot{x} \partial \dot{x}} ,\ \ \frac{\partial ^{2} L}{\partial x \partial \dot{x}} ,\ \ \frac{\partial ^{2} L}{\partial x \partial x} $$


(the derivatives are evaluated at the points of the curve $ x _{0} (t) $). It is expedient to consider the functional of $ h $ defined by (*) not only on the space $ C ^{1} $, but also on the wider space $ W _{2} ^{1} $ 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 $ A(t) $([[ Legendre condition|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 $ t _{0} $ on the interval $ (t _{0} ,\ t _{1} ) $[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)
How to Cite This Entry:
Second variation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Second_variation&oldid=44358
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article