Necessary Condition for Integral Functional to have Extremum for given Function/Non-differentiable at Intermediate Point
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Theorem
Let $y, F$ be real functions.
Let $y$ be continuously differentiable for $x \in \hointr a c \cap \hointl c b$ and satisfy:
- $\map y a = A$
- $\map y b = B$
Let $J\sqbrk y$ be a functional of the form
- $\ds J \sqbrk y = \int_a^b \map F {x, y, y'} \rd x$
Then the functional $J$ has a weak extremum if $y$ satisfies the following system of equations:
\(\ds F_y - \dfrac \d {\d x} F_{y'}\) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \bigvalueat {F_{y'} } {x \mathop = c \mathop - 0}\) | \(=\) | \(\ds \bigvalueat {F_{y'} } {x \mathop = c \mathop + 0}\) | ||||||||||||
\(\ds \bigvalueat {\paren {F - y' F_{y'} } } {x \mathop = c \mathop - 0}\) | \(=\) | \(\ds \bigvalueat {\paren {F - y' F_{y'} } } {x \mathop = c \mathop + 0}\) |
where, by the use of limit from the left and limit from the right, the following abbreviations are denoted as follows:
- $\ds \bigvalueat {\map y x} {x \mathop = c \mathop + 0} = \lim_{x \mathop \to c^+} \map y x$
- $\ds \bigvalueat {\map y x} {x \to x \mathop = c \mathop - 0} = \lim_{x \mathop \to c^-} \map y x$
The last two equations are known as the Weierstrass-Erdmann corner conditions.
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Proof
Rewrite $J \sqbrk y$ as a sum of two functionals:
\(\ds J \sqbrk y\) | \(=\) | \(\ds \int_a^b \map F {x, y, y'} \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int_a^c \map F {x, y, y'} \rd x + \int_c^b \map F {x, y, y'} \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds J_1 \sqbrk y + J_2 \sqbrk y\) |
Recall that end points $x = a,x = b$ are fixed.
The function $\map y x$ has to be $C^0$ at $x = c$, but otherwise this point can move freely.
From general variation of functional, and noting that $y = \map y x$ is an extremal, write down variations for $J_1 \sqbrk y$ and $J_2 \sqbrk y$ separately:
$\ds \delta J_1 = \bigvalueat {F_{y'} } {x \to c \mathop - 0} \delta y_1 + \bigvalueat {\paren {F - y' F_{y'} } } {x \to c \mathop - 0} \delta x_1$
$\ds \delta J_2 = \bigvalueat {-F_{y'} } {x \to c \mathop + 0} \delta y_1 - \bigvalueat {\paren {F - y' F_{y'} } } {x \to c \mathop + 0} \delta x_1$
Note that $\delta J_1$ and $\delta J_2$ involve the same increments $\delta x_1$ and $\delta y_1$.
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Since $y = \map y x$ is an extremum of $J$, we have:
\(\ds \delta J\) | \(=\) | \(\ds \delta J_1 + \delta J_2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \bigvalueat {F_{y'} } {x \mathop = c \mathop - 0} \delta y_1 + \bigvalueat {\paren {F - y' F_{y'} } } {x \mathop = c \mathop - 0} \delta x_1 - \bigvalueat {F_{y'} } {x \mathop = c \mathop + 0} \delta y_1 - \bigvalueat {\paren {F - y' F_{y'} } } {x \mathop = c \mathop + 0} \delta x_1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \bigvalueat {\paren{F_{y'} } {x \mathop = c \mathop - 0} - \bigvalueat {F_{y'} } {x \mathop = c \mathop + 0} } \delta y_1 + \paren {\bigvalueat {\paren {F - y' F_{y'} } } {x \mathop = c \mathop - 0} - \bigvalueat {\paren {F - y' F_{y'} } } {x \mathop = c \mathop + 0} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0\) |
Since $ \delta x_1$ and $ \delta y_1$ are arbitrary, both collections of terms have to vanish independently.
$\blacksquare$
Sources
- 1877: G. Erdmann: Ueber unstetige Lösungen in der Variationsrechnung. ("On discontinuous solutions in the variational calculus.") (J. Reine Angew. Math. Vol. 82: pp. 21 – 30)
- 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 3.15$: Broken Extremals. The Weierstrass-Erdmann Conditions