In the calculus of variations, the second variation extends the idea of the second derivative test to functionals. Much like for functions, at a stationary point where the first derivative is zero, the second derivative determines the nature of the stationary point; it may be negative (if the point is a maximum point), positive (if a minimum) or zero (if a saddle point). Via the second functional, it is possible to derive powerful necessary conditions for solving variational problems, such as the Legendre–Clebsch condition and the Jacobi necessary condition detailed below.
Motivation Much of the calculus of variations relies on the first variation, which is a generalization of the first derivative to a functional. An example of a class of variational problems is to find the function y {\displaystyle y} which minimizes the integral
J [ y ] = ∫ a b f ( x , y , y ′ ) d x {\displaystyle J[y]=\int _{a}^{b}f(x,y,y')dx}
on the interval [ a , b ] {\displaystyle [a,b]} ; J {\displaystyle J} here is a functional (a mapping which takes a function and returns a scalar). It is known that any smooth function y {\displaystyle y} which minimizes this functional satisfies the Euler-Lagrange equation
f y − d d x f y ′ = 0. {\displaystyle f_{y}-{\frac {d}{dx}}f_{y'}=0.}
These solutions are stationary, but there is no guarantee that they are the type of extremum desired (completely analogously to the first derivative, they may be a minimum, maximum or saddle point). A test via the second variation would ensure that the solution is a minimum.
Derivation Take an extremum y {\displaystyle y} . The Taylor series of the integrand of our variational functional about a nearby point y + ε h {\displaystyle y+\varepsilon h} where ε {\displaystyle \varepsilon } is small and h {\displaystyle h} is a smooth function which is zero at a {\displaystyle a} and b {\displaystyle b} is
f ( x , y , y ′ ) = f ( x , y , y ′ ) + ε ( h f y + h ′ f y ′ ) + ε 2 2 ( h 2 f y y + 2 h h ′ f y y ′ + h ′ 2 f y ′ y ′ ) + O ( ε 3 ) . {\displaystyle f(x,y,y')=f(x,y,y')+\varepsilon (hf_{y}+h'f_{y'})+{\frac {\varepsilon ^{2}}{2}}(h^{2}f_{yy}+2hh'f_{yy'}+h'^{2}f_{y'y'})+O(\varepsilon ^{3}).}
The first term of the series is the first variation, and the second is defined to be the second variation:
δ 2 J ( h , y ) := ∫ a b h 2 f y y + 2 h h ′ f y y ′ + f y ′ y ′ h ′ 2 . {\displaystyle \delta ^{2}J(h,y):=\int _{a}^{b}h^{2}f_{yy}+2hh'f_{yy'}+f_{y'y'}h'^{2}.}
It can then be shown that J {\displaystyle J} has a local minimum at y 0 {\displaystyle y_{0}} if it is stationary (i.e. the first variation is zero) and δ 2 J ( h , y 0 ) ≥ 0 {\displaystyle \delta ^{2}J(h,y_{0})\geq 0} for all h {\displaystyle h} .
The Jacobi necessary condition
The accessory problem and Jacobi differential equation
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