CDS 140b Spring 2014 Homework 1

R. Murray, D. MacMartin	Issued: 2 Apr 2014 (Wed)
CDS 140b, Spring 2014	Due: 9 Apr 2014 (Wed)

__MATHJAX__

Note: In the upper left hand corner of the second page of your homework set, please put the number of hours that you spent on this homework set (including reading).

1. Perko, Section 2.14, problem 1
   (a) Show that the system <amsmath>\aligned

   \dot x&=a_{11}x+a_{12}y+Ax^2-2Bxy+Cy^2\\ \dot y&=a_{21}x-a_{11}y+Dx^2-2Axy+By^2

   \endaligned</amsmath>

   is a Hamiltonian system with one degree of freedom; i.e., find the Hamiltonian function $H(x,y)$ for this system.
   (b) Given $f\in C^2(E)$, where $E$ is an open, simply connected subset of $\mathbb R^2$, show that the system $\dot{x}=f(x)$ is a Hamiltonian system on $E$ iff $\nabla\cdot f(x)=0$ for all $x\in E$.

2. Perko, Section 2.14, problem 7. Show that if $x_0$ is a strict local minimum of $V(x)$ then the function $V(x)-V(x_0)$ is a strict Lyapunov function (i.e., $\dot{V}<0$ for $x\neq 0$) for the gradient system $\dot x=-\mathrm{grad}V(x)$.
3. Perko, Section 2.14, problem 12. Show that the flow defined by a Hamiltonian system with one degree of freedom is area preserving. Hint: Cf. Problem 6 in Section 2.3
4. A planar pendulum (in the $x$-$z$ plane) of mass $m$ and length $\ell$ hangs from a support point that moves according to $x=a\cos (\omega t)$. Find the Lagrangian, the Hamiltonian, and write the first-order equations of motion for the pendulum.