Conditions for Strong Minimum of Functional
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Theorem
Let $\mathbf y$ be an $n$-dimensional vector such that:
- $\map {\mathbf y} a = A$ and $\map {\mathbf y} b = B$
Let $J$ be a functional such that:
- $\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$
Let $\gamma$ be an extremal curve of $J$.
Let the following be the field of the functional $J$:
- $\mathbf y' = \map {\boldsymbol \psi} {x, \mathbf y}$
Let $R$ be an open region containing $\gamma$ and have the field $\boldsymbol \psi$ defined as $\forall \paren {x, \mathbf y} \in R$.
Let $\mathbf w$ be a finite vector.
Suppose that:
- $\forall \paren {x, \mathbf y} \in R: \map E {x, \mathbf y, \boldsymbol \psi, \mathbf w}\ge 0$
where $E$ is Weierstrass E-Function.
Then $J$ has a strong minimum for $\gamma$.
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Proof
By definition, the increment of $J$ is:
- $\ds \Delta J = \int_{\gamma^*} \map F {x, \mathbf y, \mathbf y'} \rd x - \int_\gamma \map F {x, \mathbf y, \mathbf y'} \rd x$
where $\gamma$ and $\gamma^*$ are curves described by $\paren {x, \map {\mathbf y} x}$ and $\paren {x, \paren {\mathbf y^*} x}$ respectively, such that:
- $\map {\mathbf y^*} x - \map {\mathbf y} x = \map {\mathbf h} x$
Consider Hilbert's invariant integral:
\(\ds \map g {x, \mathbf y}\) | \(=\) | \(\ds \int_\Gamma \paren {-H \rd x + \mathbf p \rd \mathbf y}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int_\Gamma \paren {-\paren {-F + \mathbf y' F_{\mathbf y'} } \rd x + F_{\mathbf y'} \rd \mathbf y}\) | Definition of Hamiltonian and Momentum | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_\Gamma \paren {F \rd x - \mathbf y' F_{\mathbf y'} \rd x + F_{\mathbf y'} \rd \mathbf y}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int_{\Gamma} \paren {F \rd x - \boldsymbol \psi F_{\mathbf y'} \rd x + F_{\mathbf y'} \rd \mathbf y}\) | $\mathbf y' = \map {\boldsymbol \psi} {x, \mathbf y}$ |
Since the integrand is full differential, the integral does not depend on the shape of $ \Gamma $, but only on its endpoints.
Therefore, for $\Gamma = \gamma$ and $\Gamma = \gamma^*$ the value of the integral is the same.
Since $\map {\mathbf y'} x = \map {\boldsymbol \psi} {x, \mathbf y}$ determines boundary conditions for $\gamma$, $\Gamma = \gamma$ is one of the trajectories of the field $\map {\mathbf y'} x = \map {\boldsymbol \psi} {x, \mathbf y}$.
Hence, $\d \mathbf y$ is constrained by $\d \mathbf y = \boldsymbol \psi \rd x$:
- $\ds \map g {x, \mathbf y} = \int_\gamma \map F {x, \mathbf y, \mathbf y'} \rd x$
Thus, $\map g {x, \mathbf y'}$ can be written in two different ways.
- $\ds \int_\gamma \map F {x, \mathbf y, \mathbf y'} \rd x = \int_{\gamma^*} \paren {\paren {\map F {x, \mathbf y, \boldsymbol \psi} - \boldsymbol \psi F_{\mathbf y'} \paren {x, \mathbf y, \boldsymbol \psi} } \rd x + F_{\mathbf y'} \paren {x, \mathbf y, \boldsymbol \psi} \rd \mathbf y}$
Substitute this into the expression for $\Delta J$:
\(\ds \Delta J\) | \(=\) | \(\ds \int_{\gamma^*} \map F {x, \mathbf y, \mathbf y'} \rd x - \int_{\gamma^*} \paren {\paren {\map F {x, \mathbf y, \boldsymbol \psi} - \boldsymbol \psi \map {F_{\mathbf y'} } {x, \mathbf y, \boldsymbol \psi} }\rd x + \map {F_{\mathbf y'} } {x, \mathbf y, \boldsymbol \psi} \rd \mathbf y}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int_{\gamma^*} \paren {\map F {x, \mathbf y, \mathbf y'} - \map F {x, \mathbf y, \boldsymbol \psi} - \paren {\mathbf y'-\boldsymbol \psi} \map {F_{\mathbf y'} } {x, \mathbf y, \boldsymbol \psi} }\rd x\) | as $\d \mathbf y = \boldsymbol \psi \rd x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_{\gamma^*} \map E {x, \mathbf y, \boldsymbol\psi, \mathbf y'} \rd x\) | Definition of Weierstrass E-Function |
By assumption:
- $\map E {x, \mathbf y, \boldsymbol \psi, \mathbf y'} \ge 0$
Hence the integrand is bounded below by $0$ and above by its maximum in the interval of integration.
Hence, the integral is bounded below by $0$ and above by some positive number.
Therefore:
- $\Delta J \ge 0$
$\blacksquare$
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Sources
- 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 6.34$: The Weierstrass E-Function. Sufficient Conditions for a Strong Extremum