CDS 140b Spring 2014 Homework 2 - Revision history

Macmardg at 16:34, 7 April 2014

2014-04-07T16:34:53Z

← Older revision		Revision as of 16:34, 7 April 2014
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	where the potential function		where the potential function
	<center><amsmath>		<center><amsmath>
	\Omega(x,y)=\frac{x^2+y^2}{2}+\frac{1-\mu}{r_1}+\frac{\mu}{r_2}+\frac{\mu(1-\mu){2}		\Omega(x,y)=\frac{x^2+y^2}{2}+\frac{1-\mu}{r_1}+\frac{\mu}{r_2}+\frac{\mu(1-\mu)}{2}
	</amsmath></center>		</amsmath></center>
	with $r_1=\sqrt{(x+\mu)^2+y^2}$ and $r_2=\sqrt{(x-1+\mu)^2+y^2}$.		with $r_1=\sqrt{(x+\mu)^2+y^2}$ and $r_2=\sqrt{(x-1+\mu)^2+y^2}$.

Macmardg at 16:33, 7 April 2014

2014-04-07T16:33:51Z

← Older revision		Revision as of 16:33, 7 April 2014
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	H=\frac{(p_x+y)^2+(p_y-x)^2}{2}+\Omega(x,y)		H=\frac{(p_x+y)^2+(p_y-x)^2}{2}+\Omega(x,y)
	</amsmath></center>		</amsmath></center>
			where the potential function
			<center><amsmath>
			\Omega(x,y)=\frac{x^2+y^2}{2}+\frac{1-\mu}{r_1}+\frac{\mu}{r_2}+\frac{\mu(1-\mu){2}
			</amsmath></center>
			with $r_1=\sqrt{(x+\mu)^2+y^2}$ and $r_2=\sqrt{(x-1+\mu)^2+y^2}$.
	Show that the equilibrium points are given by the critical points of the (messy) function $\Omega$.		Show that the equilibrium points are given by the critical points of the (messy) function $\Omega$.
	(This leads to 5 solutions, for L1 through L5, with L1, L2, and L3 collinear, i.e., y=0.)		(This leads to 5 solutions, for L1 through L5, with L1, L2, and L3 collinear, i.e., y=0. If you are interested, the collinear solutions are not too difficult to solve for numerically for some $\mu$.)
	To explore the linearized dynamics it is sufficient to retain only quadratic terms in $H$ (why?). For the collinear Lagrange points this leads to		To explore the linearized dynamics it is sufficient to retain only quadratic terms in $H$ (why?). For the collinear Lagrange points this leads to
	<center><amsmath>		<center><amsmath>

Macmardg at 00:06, 6 April 2014

2014-04-06T00:06:58Z

@@ Line 31: / Line 31: @@
 </li>
 <li>'''Perko, Section 3.6, problem 4.'''
 </li>
 </ol>

Murray at 18:57, 31 March 2014

2014-03-31T18:57:25Z

← Older revision		Revision as of 18:57, 31 March 2014
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	this homework set (including reading).		this homework set (including reading).

	~~'''~~UNDER CONSTRUCTION, DO NOT START~~'''~~		{{warning\|UNDER CONSTRUCTION, DO NOT START}}

	<ol>		<ol>

Macmardg at 22:42, 29 March 2014

2014-03-29T22:42:54Z

← Older revision		Revision as of 22:42, 29 March 2014
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	<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
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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.)		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>

Macmardg: 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..."

2014-03-25T22:27:18Z

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..."

New page

{{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 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'''

<ol>
<li>'''Perko, Section 3.3, problem 5.'''
Show that
<center><amsmath>\aligned
\dot x &=y+y(x^2+y^2)\\
\dot y &=x-x(x^2+y^2)
\endaligned</amsmath></center>
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
<center><amsmath>
(x^2+y^2)^2-2(x^2-y^2)=C
</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.)
</li>
</ol>