Definition:Invariant Functional under Transformation

From ProofWiki
Jump to navigation Jump to search


Let $y_i$, $F$, $\Phi$, $\Psi$ be real functions.

Let $\mathbf y = \sequence {y_i}_{1 \mathop \le i \mathop \le n}$.

Let $\ds J \sqbrk {\mathbf y} = \int_{x_0}^{x_1} \map F {x, \mathbf y, \mathbf y'} \rd x$ be a functional.


$X = \map \Phi {x, \mathbf y, \mathbf y'}$
$\mathbf Y = \map {\mathbf \Psi} {x, \mathbf y, \mathbf y'}$

Let curve $\gamma$ defined by:

$\mathbf y = \map {\mathbf y} x, \quad x_0 \le x \le x_1$

be transformed into a curve $\Gamma$ defined by:

$\mathbf Y = \map {\mathbf Y} X, \quad X_0 \le X \le X_1$

Then the functional $J \sqbrk {\mathbf y}$ is invariant under the given transformation if and only if:

$J \sqbrk \Gamma = J \sqbrk \gamma$

That is, if and only if:

$\displaystyle \int_{x_0}^{x_1} \map F {x, \mathbf y, \mathbf y'} \rd x = \int_{X_0}^{X_1} \map F {X, \mathbf Y, \mathbf Y'} \rd X$