Variational calculus

Essentially the mathematics of optimizing non-parametrized functions in function spaces. Variational calculus uses the Gâteux derivative of functionals: $$\boxed{\mathcal{F}[f;g]=:=\frac{d}{dh}\mathcal{F}[f+hg]|_{h=0}},$$which we can think of as an arbitrary "directional" derivative of the functional $\mathcal{F}$, evaluated at $f$, in the direction of $g$ with the infinitesmal perturbation size $h$.

Important class of functionals -> Euler-Lagrange equations

The functional (map from functions to $\mathbb{R}$) $$S[y]=\int_{a}^bF(x,y,y')dx,\quad y(a)=A,\quad y(b)=B,\quad y'=\frac{dy}{dx}$$has stationary paths given by the Euler-Lagrange equations:$$\frac{\partial F}{\partial y}-\frac{d}{dx}\left(\frac{\partial F}{\partial y'} \right)=0$$

Beltrami identity

If $L$ doesn't depend explicitly on $x$, then, we can prove that:$$y'\frac{\partial F}{\partial y'} - F = \text{const}$$
Proof: