stevengogogo
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1 month ago Variational Calculus
- Variational calculus uses small changes in functions to find maxima and minima of function.
- Functionals are expressed as definite integrals involving functions and their derivatives.
- Extrema can be found using Euler-Lagrange equation
- functional: map functions to scalars
- Extrema of functional $J[y]$ if $\Delta J = J[y] - J[f]$ has the same sign for all y in small neighborhood of $f$
- weak extrema: first derivative of continuous function are not continuous.
- A necessary condition to find weak extrema is Euler-Lagrange Equation
Euler-Lagrange Equation
Find extrema of functional
- Extrema of functional obtained by finding functional for that the functional derivative is zero. => by solving Euler-Lagrange Equation
Example of the functional
$$ J[y] = \int^{x2}{x_1} L(x,y(x), y'(x))dx $$
where
- $x_1, x_2$: constants
- $y(x)$ is twice continuous differentiable
- $y'(x) = \frac{dy}{dx}$
- $L(x, y(x), y'(x))$ is twice continuous differentiable with respect to x, y, y'
Local minimum at $f$
$$ J[f] \leq J[f+\epsilon \eta] $$
- $\eta(x)$: an arbitrary function at least one derivative and vanishes at endpoints $x_1$, $x_2$
- $\epsilon$ close to $0$
- $\epsilon \eta$ is variation of the funtion $f$ denoted by $\delta f$
Let
$$ \phi(\epsilon) = J[f+ \epsilon \eta] $$
Since $J[y]=J[f+\epsilon\eta]$ has a minimum for $y=f$,
$$ \phi'(0) = \frac{d\phi}{d\epsilon}_{\epsilon=0} =0 $$
Total derivative of $L[x,y,y']$ where $y=f+\epsilon \eta(x)$
Good learning material about Calculus of variations:
https://en.wikipedia.org/wiki/Calculus_of_variations