Problem 1.

(1) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a strictly convex $C^{2}$ function and let
$$
F[u(\cdot)]=\int_{-\frac{1}{3}}^{0} f(u(3 y+1)) d y .
$$
Consider the problem:
minimize $F[u(\cdot)]$
subject to: $u \in \mathcal{A}$,
where $\mathcal{A}:=\left{u:[0,1] \rightarrow \mathbb{R} \mid u \in C^{1}[0,1]\right.$ and $\left.u(0)=0 \quad u(1)=1\right}$.
(a) Find the Euler-Lagrange equation. Hint: you would first need to make a substitution to express $F[u(\cdot)]$ in an appropriate form.
(b) Solve the Euler-Lagrange equation. Hint: use the convexity assumption.

Problem 2. (2) Let
$$
F[x(\cdot), y(\cdot), z(\cdot)]=\int_{0}^{1}\left[\dot{x}(t)^{2}+\dot{y}(t)^{2}+\dot{z}(t)^{2}\right]^{1 / 2} d t
$$
and
$$
H[x, y, z]=x^{2}+y^{2}+z^{2}-1=0 .
$$
(a) Find (but don’t solve) the Euler-Lagrange equations for the functional $F$ subject to the holonomic constraint $H$.
(b) Show that the curve $x(t)=\sin t \cos \alpha, y(t)=\sin t \sin \alpha, z(t)=$ $\cos t$ solves the Euler-Lagrange equations you found in part (a). Here $\alpha$ is fixed.

Problem 3. (3) Let
$$
F[x(\cdot), y(\cdot)]=\int_{0}^{1}\left{\dot{x}(t)^{2}+\dot{y}(t) x(t)\right} d t
$$
and
$$
H[x(t), y(t)]=x(t)^{2}-y(t)=0 .
$$
(a) Find the Euler-Lagrange equations for the holonomic problem above.
(b) Solve the Euler-Lagrange equations.

(4) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a strictly convex $C^{2}$ function. Let
$$
F[u(\cdot)]=\int_{0}^{1} f(u(x)) d x,
$$
and
$$
\left.G[u(\cdot)]=\int_{0}^{1} u(x)\right) d x
$$
Consider the problem:
$$
\begin{aligned}
\text { minimize } & F[u(\cdot)] \
\text { subject to: } & G[u(\cdot)]=5 \
& u \in \mathcal{A}
\end{aligned}
$$
where $\mathcal{A}:=\left{u:[0,1] \rightarrow \mathbb{R} \mid u \in C^{1}[0,1]\right.$ and $\left.u(0)=0 \quad u(1)=1\right}$.
(a) Find the Euler-Lagrange equation.
(b) Solve the Euler-Lagrange equation.
The last two problems are suggested for practice and are not to be turned in.
(5) Let $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ be a $C^{1}$ function and consider the functional:
$$
F[u(\cdot)]=f\left(\int_{a}^{b} \delta_{1}(x) u(x) d x, \ldots, \int_{a}^{b} \delta_{n}(x) u(x) d x\right)
$$
where for each $i=1, \ldots, n, \delta_{i}(x)$ is 1 for $x \in\left[a+(i-1) \frac{b-a}{n}, a+i \frac{b-a}{n}\right]$ and 0 elsewhere. As usual the space $\mathcal{A}={u:[a, b] \rightarrow \mathbb{R} \mid u \in$ $\left.C^{1}, u(a)=A, u(b)=B\right}$. Find the first order condition for a minimizer $u_{}(\cdot)$ of $F$ in $\mathcal{A}$. Hint: start by computing the “directional derivative” $0=\left.\frac{d}{d s}\right|{s=0} F\left[u{}(\cdot)+s v(\cdot)\right]=\cdots$, where $v(\cdot)$ is a test function.
(6) Let $\mathcal{A}:=\left{\mathbf{u}=\left(u_{1}, u_{2}, u_{3}\right):[0,1] \rightarrow \mathbb{R}^{3} \mid \mathbf{u} \in C^{1}\right}$ and consider the holonomic problem:
minimize $F[\mathbf{u}(\cdot)]:=\int_{0}^{1} \sqrt{\dot{u}{1}(t)^{2}+\dot{u}{2}(t)^{2}+\dot{u}{3}(t)^{2}} d t$ subject to: $\mathbf{u} \in \mathcal{A}, \quad G\left(u{1}(t), u_{2}(t), u_{3}(t)\right):=u_{1}(t)+u_{2}(t)-1 \equiv 0$.
Find, but do not solve, the Euler-Lagrange equations for this problem.