Second variation

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)