CDS 140b Spring 2014 Homework 2: Difference between revisions
(Created page with " {{CDS homework | instructor = R. Murray, D. MacMartin | course = CDS 140b | semester = Spring 2014 | title = Problem Set #2 | issued = 9 Feb 2014 (Wed) | due = 16 Feb 2...") |
No edit summary |
||
| Line 15: | Line 15: | ||
<ol> | <ol> | ||
<li> | |||
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. | |||
</li> | |||
<li>'''Perko, Section 3.3, problem 5.''' | <li>'''Perko, Section 3.3, problem 5.''' | ||
Show that | Show that | ||
| Line 25: | Line 28: | ||
(x^2+y^2)^2-2(x^2-y^2)=C | (x^2+y^2)^2-2(x^2-y^2)=C | ||
</amsmath></center> | </amsmath></center> | ||
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 | 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, noting the occurrence of a compound separatrix cycle. | ||
</li> | |||
<li>'''Perko, Section 3.6, problem 4.''' | |||
</li> | </li> | ||
</ol> | </ol> | ||
Revision as of 22:42, 29 March 2014
| R. Murray, D. MacMartin | Issued: 9 Feb 2014 (Wed) |
| CDS 140b, Spring 2014 | Due: 16 Feb 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).
UNDER CONSTRUCTION, DO NOT START
- 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.
- 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, noting the occurrence of a compound separatrix cycle.
- Perko, Section 3.6, problem 4.
