Difference between revisions of "CDS 140b Spring 2014 Homework 2"

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(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...")
 
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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.
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<li>'''Perko, Section 3.3, problem 5.'''
 
<li>'''Perko, Section 3.3, problem 5.'''
 
Show that
 
Show that
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(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.  (You need not comment on the compound separatrix cycle.)
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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.
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<li>'''Perko, Section 3.6, problem 4.'''
 
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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

  1. 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.
  2. 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.

  3. Perko, Section 3.6, problem 4.