# CDS 140b Spring 2014 Homework 1

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R. Murray, D. MacMartin | Issued: 2 Apr 2014 (Wed) |

CDS 140b, Spring 2014 | Due: 9 Apr 2014 (Wed) |

**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).

**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 $x\in E$.-
**Perko, Section 2.14, problem 4.**Given the function $U(x)$ in the text, sketch the phase portrait for the Newtonian system with Hamiltonian $H(x,y)=y^2/2+U(x)$ -
**Perko, Section 2.14, problem 5 a,b,c.**

For each of the following Hamiltonian functions, sketch the phase portraits for the Hamiltonian system and the gradient system orthogonal to it. Draw both phase portraits on the same phase plane.

(a) $H(x,y)=x^2+2y^2$

(b) $H(x,y)=x^2-y^2$

(c) $H(x,y)=y\sin{x}$ -
**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 $\dot x=-\mathrm{grad}V(x)$. **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- 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.