这是一份多伦多大学的Analytical mechanics APM462: Homework 6作业代写成功案例
Analytical mechanics
(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.
$$
F[x(\cdot), y(\cdot), z(\cdot)]=\int_{0}^{1}\left[\dot{x}(t)^{2}+\dot{y}(t)^{2}+\dot{z}(t)^{2}\right]^{1 / 2} d t
$$
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.
$$
F[x(\cdot), y(\cdot)]=\int_{0}^{1}\left{\dot{x}(t)^{2}+\dot{y}(t) x(t)\right} d t
$$
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.
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- Lectures: Mon 6-9pm in LM 162
- Office hours begin the week of Jan 15.
- The suggested textbook for this course is Linear and Non-Linear Programming (4th Edition) by David Luenberger & Yinyu Ye. Publisher: Springer. Should be available at the UT bookstore. The textbook is suggested and not required: I will try to make the lectures and HW self contained.Before classes begin, I suggest reviewing some multivariable calculus, especially the part about partial derivatives, gradients, Taylor expansion, and the Jacobian matrix. Refreshing basic linear algebra would also be helpful.
- Course Syllabus (approximate):
- Linear and Non-Linear Programming: parts of chapters 7,8,9,11.
- The Calculus of Variations.
- Supplementary material from lectures.
- The Course Grade will be calculated as follows:
- There will be 5 or 6 homework assignments. Assignments are to be handed in on assigned dates at the beginning of class.
- Please staple your homeworks. Unstapled homeworks will be penalized 2 points.
- NO late homework will be accepted.
- Important: There will be no make up term test! If you have a valid reason for missing the term test, the corresponding portion of the final exam will count as your term test grade.
- Exam dates:
- Term Test: Mon, Feb 26.
- Final Exam: Mon, Apr 30, 9am-12.
- Please read the information in the two links below about academic misconduct: Plagiarism, Misconduct.
- For more information about this course please see Blackboard.
Course Calendar (tentative)
# | Week of … | |
Winter Semester: | ||
1 | Jan 8 | Review. Finite dimentional optimization (unconstrained problems): 1st and 2nd order neccessary conditions for a minimum. |
2 | Jan 15 | Finite dimentional optimization (unconstrained problems): 2nd order sufficient condition for a minimum. Convex functions: C 1 and C 2 characterizations. |
3 | Jan 22 | Convex functions: local minimum is a global minimum, maxumum is attained on boundary of compact convex domain. Introduction to Finite dimentional optimization (equality constraints): Lagrange multipliers. |
4 | Jan 29 | Finite dimentional optimization (equality constraints): 1st and 2nd order neccessary conditions for a local minimum. 2nd order sufficient condition for a local minimum. |
5 | Feb 5 | Finite dimentional optimization (inequality constraints): 1st and 2nd order neccessary conditions for a local minimum. 2nd order sufficient condition for a local minimum. |
6 | Feb 12 | Algorithems: Newton’s method, method of steepest descent. |
Feb 19 | Reading Week | |
7 | Feb 26 | Midterm. Steepest descent. |
8 | Mar 1 | Conjugate direction methods. Conjugate gradient method. |
9 | Mar 8 | Global convergence theorem. Calculus of Variations: introduction. |
10 | Mar 15 | Calculus of Variations: 1st order necc. conditions, Euler-Lagrange equation. |
11 | Mar 22 | Calculus of Variations: Examples, classical mechanics (least action principle). |
12 | Mar 29 | Calculus of Variations: equality constraints, sufficient conditions (convexity). |
Apr 9-30 | Final Exams period |