CDS 140a Winter 2013 Homework 6
R. Murray, D. MacMartin | Issued: 12 Feb 2013 (Tue) |
ACM 101/AM 125b/CDS 140a, Winter 2013 | Due: 19 Feb 2013 (Tue) |
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
- 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.
- 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 $x\in E$.
- Show that the system
- 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
- Perko, Section 3.3, problem 5.
Show that
<amsmath>\aligned \dot x &=y+y(x^2+y^2)\\ \dot y &=x-x(x^2+y^2)
\endaligned</amsmath>is a Hamiltonian system with $4H(x,y)=(x^2+y^2)^2-2(x^2-y^2)$. Show that $dH/dt=0$ along solution curves of this system and therefore that solution curves of this system are given by
<amsmath> (x^2+y^2)^2-2(x^2-y^2)=C
</amsmath>Show that the origin is a saddle for this system and that $(\pm 1,0)$ are centers for this system. (Note the symmetry with respect to the $x$-axis.) Sketch the two homoclinic orbits corresponding to $C=0$ and sketch the phase portrait for this system. (You need not comment on the compound separatrix cycle.)
- 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.