Jacobi's Equation is Variational Equation of Euler's Equation

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Theorem

The Variational equation of Euler's equation is Jacobi's equation.

Proof

Let Euler's equation be

$\displaystyle \map {F_y} {x,\hat y,\hat y'}-\frac \d {\d x} \map {F_{y'} } {x,\hat y,\hat y'}= 0$

which is derived from:

$\displaystyle\int_a^b \paren {\displaystyle\map {F_y} {x,\hat y,\hat y'}-\frac \d {\d x} \map {F_{y'} } {x,\hat y,\hat y'} } \rd x =0$

Let $\map {\hat y} x=\map y x$ and $\map {\hat y} x=\map y x+\map h x$ be solutions of Euler's equation.

By Taylor's theorem:

\(\displaystyle \map {F_y} {x,y+h,y'+h'}-\frac \d {\d x} \map {F_{y'} } {x,y+h,y'+h'}\) \(=\) \(\displaystyle F_y+F_{yy}h+F_{yy'}h'+\map {\mathcal O} {h^2,h'^2}-\frac \d {\d x} \paren { F_{y'}+F_{y'y}h+F_{y'y'}h'+\map {\mathcal O} {h^2,h'^2} }\)
\(\displaystyle \) \(=\) \(\displaystyle \sqbrk {F_y-\frac \d {\d x} F_{y'} }+F_{yy}h+\map {\mathcal O} {h^2,hh',h'^2}-F_{y'y}'h-F_{y'y'}' h'-F_{y'y'}h''-\frac \d {\d x} \map {\mathcal O} {h^2,hh',h'^2}\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {F_{yy}-\frac \d {\d x}F_{y'y} }h-\frac \d {\d x} \paren {F_{y'y'}h'} +\map {\mathcal O} {h^2,hh',h'^2}\)

where the ommited ordered set of variables is $\paren{x,y,y'}$, and $\map {\hat y} x=\map y x$ has been used as a solution to $\displaystyle F_y-\frac \d {\d x}F_{y'}=0 $.

Therefore, Euler's equation is to be derived from

$\displaystyle\int_a^b\paren { \paren { F_{yy}-\frac \d {\d x}F_{y'y} }h-\frac \d {\d x} \paren { F_{y'y'}h'}+\map {\mathcal O} {h^2,hh',h'^2} }\rd x=0$

By integration by parts,

$\displaystyle\int_a^b\map {\mathcal O} {h^2,hh',h'^2}\rd x=\int_a^b\map {\mathcal O} {h^2}\rd x$

Thus, the equivalent differential equation is

$\displaystyle\paren {F_{yy}-\frac \d {\d x}F_{y'y} }h-\frac \d {\d x} \paren { F_{y'y'}h'}+\map {\mathcal O} {h^2}=0$

Omission of $\map {\mathcal O} {h^2}$ and multiplication of equation by $\frac 1 2$ yields Jacobi's equation.


$\blacksquare$


Sources

1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 5.27$: Jacobi's Necessary Condition. More on Conjugate Points